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33  WEST  MAIN  STREET 

WEBSTER.  N.Y.  MS80 

(716)  873-4503 


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CIHM/ICMH 

Microfiche 

Series. 


CIHIVI/ICMH 
Collection  de 
microfiches. 


Canadian  Institute  for  Historical  Microreproductions  /  Institut  Canadian  de  microreproductions  historiques 


Technical  and  Bibliographic  Notes/Notes  techniques  et  bibliographiques 


The  Institute  has  attempted  to  obtain  the  best 
original  copy  available  for  filming.  Features  of  this 
copy  which  may  be  bibliographically  unique, 
which  may  alter  any  of  the  images  in  the 
reproduction,  or  which  may  significantly  change 
the  usual  method  of  filming,  are  checked  below. 


L'Institut  a  microfilm^  le  meilleur  exemplaire 
qu'il  lui  a  M  possible  de  se  procurer.  Let  details 
de  cet  exemplaire  qui  sont  peut-4tre  uniques  du 
point  de  vue  bibliographique,  qui  peuvent  modifier 
une  image  reproduite,  ou  qui  peuvent  exigtir  une 
modification  dans  la  m6thode  normale  de  filmage 
sont  indiquAs  ci-dessous. 


The 
tot 


The 
pes 
oft 
film 


□    Coloured  covers/ 
Couverture  de  couleur 


I      I    Coloured  pages/ 


□ 
D 
D 


□ 
□ 


./ 


7f 


□ 


Covers  damaged/ 
Couverture  endommagde 

Covers  restored  and/or  laminated/ 
Couverture  restaurie  et/ou  pelliculAe 

Cover  title  missing/ 

Le  titre  de  couverture  manque 

Coloured  maps/ 

Cartes  gdographiques  en  couleur 

Coloured  ink  (i.e.  other  than  blue  or  black)/ 
Encre  de  couleur  (i.e,.  autre  que  bleue  ou  noirel 

Coloured  plates  and/or  illustrations/ 
Planches  et/ou  illustrations  en  couleur 

Bound  with  other  material/ 
Reli^  avec  d'autres  documents 

Tight  binding  may  cause  shadows  or  distortion 
along  interior  margin/ 

La  reliure  serree  peut  causer  de  I'ombre  ou  de  la 
distortion  le  long  de  la  marge  intirieure 

Blank  leaves  added  during  restoration  may 
appear  within  the  text.  Whenever  possible,  these 
have  been  omitted  from  filming/ 
II  se  peut  que  certaines  pages  blanches  ajouties 
lors  d'une  restauration  apparaissent  dans  le  texte. 
mais,  lorsque  cela  itait  possible,  ces  pages  n'ont 
pas  Ct6  film^es. 


Pages  de  couleur 

Pages  damaged/ 
Pages  endommag^es 


D 
0^ 


Pages  restored  and/or  laminated/ 
Pages  restaur6es  et/ou  pelliculies 

Pages  discoloured,  stained  or  foxed/ 
Pages  d6color6es,  tachet6es  ou  piqu^es 


□    Pages  detached/ 
Pages  ditachdes 

r~T'  Showthrough/ 
I — I    Transparence 


D 
D 
D 
D 


Quality  of  print  varies/ 
Quality  inigale  de  I'impression 

Includes  supplementary  material/ 
Comprend  du  materiel  supplementaire 

Only  edition  available/ 
Seule  Edition  disponible 

Pages  wholly  or  partially  obscured  by  errata 
slips,  tissues,  etc.,  have  been  refilmed  to 
ensure  the  best  possible  image/ 
Las  pages  totalement  ou  partiellement 
obscurcies  par  un  feuillet  d'errata,  une  pelure, 
etc.,  ont  4t6  filmies  ^  nouveau  de  fapon  A 
obtenir  la  meilleure  imoge  possible. 


Ori( 
beg 
the 
sioi 
othi 
first 
sior 
or  11 


Th« 
she 
TIN 
whi 

Mai 
diff 
enti 
beg 
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met 


n 


Additional  comments:/ 
Commentaires  supplAmentaires; 


This  item  is  filmed  at  the  reduction  ratio  checked  below/ 

Ce  document  est  film^  au  taux  de  reduction  indiquA  ci-dessous. 

10X  14X  18X  22X 


28X 


30X 


J 

12X 


16X 


aox 


24X 


28X 


32X 


Th«  copy  film«d  h«r«  has  b««n  raproducad  thanks 
to  tha  ganarosity  of: 

UniMfiity  of  British  Columbia  Library 


L'axamplaira  film*  fut  raproduit  grica  it  la 
gintrosit*  da: 

University  of  British  Columbia  Library 


Tha  imvgas  appaaring  hara  era  tha  bast  quality 
possibia  considaring  tha  condition  and  lagibility 
of  tha  original  copy  and  in  kaaping  with  tha 
filming  contract  spacifications. 


Las  ir^iagas  suivantas  ont  AtA  raproduitas  avac  la 
plus  grand  join,  compta  tanu  da  ia  condition  at 
da  la  nattat*  da  l'axamplaira  filmA,  at  an 
con'/ormit*  avac  las  conditions  du  contrat  da 
filmaga. 


Original  copias  in  printad  papar  covars  ara  filmad 
baginning  with  tha  front  covar  and  anding  on 
tha  last  paga  with  a  printad  or  illustratad  impras- 
sion.  or  tha  back  covar  whan  appropriata.  All 
othar  original  copias  are  filmad  baginning  on  tha 
first  paga  with  a  printad  or  illustratad  impras- 
sion.  and  anding  on  tha  last  paga  with  a  printad 
or  illustratad  imprassion. 


Las  cxamplairas  orig!naux  dont  la  couvartura  Bn 
papiar  ast  imprimte  sont  filmAs  an  commandant 
par  la  pramiar  plat  at  an  tarminant  soit  par  la 
darniira  paga  qui  comporta  una  amprainta 
d'imprassion  ou  d'illustratlon,  soit  par  la  sacond 
plat,  salon  la  cas.  Tous  las  autras  axamplairas 
originaux  sont  filmAs  en  commandant  par  la 
pramiire  rage  qui  comporte  une  empreinte 
d'impression  ou  d'illustration  at  en  termlnant  par 
la  derniAre  page  qui  comporte  une  telle 
empreinte. 


The  last  recorded  frame  on  each  microfiche 
shall  contain  tha  symbol  -^^  (meaning  "CON- 
TINUED ").  or  the  symbol  V  (meaning  "END  "), 
whichever  applies. 


Un  des  symboles  suivants  apparaitra  sur  la 
darniire  image  de  cheque  microfiche,  selon  le 
cas:  le  symbole  — »■  signifie  "A  SUIVRE",  le 
symbole  V  signifie  "FIN". 


Maps,  plates,  charts,  etc..  may  be  filmed  at 
different  reduction  ratios.  Those  too  large  to  be 
entirely  included  in  one  exposure  ara  filmad 
beginning  in  the  upper  left  hand  corner,  left  to 
right  and  top  to  bottom,  as  many  frames  as 
required.  The  following  diagrams  illustrate  the 
method: 


Lea  cartes,  planches,  tableaux,  etc.,  peuvent  Atre 
fiimis  A  des  taux  de  rMuction  diffirents. 
Lorsque  le  document  est  trop  grand  pour  Atre 
reproduit  en  un  seul  clichA,  il  est  film*  A  partir 
de  Tangle  supArieur  gauche,  de  gauche  A  droite, 
et  de  haut  en  bas,  en  prenant  le  nombre 
d'imagas  nAcessaire.  Les  diagrammes  suivants 
illustrant  la  mAthode. 


i 

t 

3 

1 

2 

3 

4 

5 

6 

}\ 


THEORETICAL  ASTRONOMY 


KELATI.\(i    TO   TUB 


Motions  of  the  Heavenly  Bodies 


REVOLVING  AROJ-ND  THE  SUN  IN  ACmRDANfE  WITH 
THE  LAW  OF  UNIVEllSAL  GllAVITATlUiN 


KjinRAcrso 

A   "T.-TEMATIf  DEHn-ATtON  OF  THE  FOnMt'L.*;  FOK  THE  lAUlTUTION    OF   TIIF.  OEOiEVTIlli'   *VD  IIIUO- 

rKNTKt      ll.A   K-.  rim  TIIK   DKIKllJIIXATKlN  OK  THE  ORniT^  ilK  IM.WKrs   AMI  <  iiMEI'i,  tuR 

THE  iiiRRrrlOX   ilK   AIM'IlilXIMVTK   KI.KMf.MS    AVJi  IMH  THK   i  ivMi  IT  \Tjn>   OF 

SI'E'IAI.  1  ERTLRIIATION'.:  TOilETIlEK  WITH  THE  THK.ollV  iiK  THK  inMUl- 

NATION  or  UIISERVATIUN.-!  AND  THE  SaiUOD  UF  LKAiiT  tliUAKS*. 


ttulith  Jlumcrical  Oh'amjjlcs  and  Suxiliarg  3'ablfs 


BY 

JAMES    V.   WATSON 

lllREl'TOIt  OF  THE  ODSERVATOKV  AT   AW   ARnoil    A\n  PROFESSOR  OF  ASTBOXOJIT  IN 
VMVER.<1TY  OF  MH  UIUAN 


PHTT-AT>FTiPIITA 

J.  B.  LTPPINCOTT    &    CO. 

LONDON:   TRUnNER  <fe  CO. 

18G8 


A 


KiiloriMl,  iK'cordliiK  to  Ai'l  of  ( 'oiiKri'st.H,  in  tlu'  yi'lir  imw,  liy 
.1.  II.   LlPl'I  NCOTT    A    CO., 

Ill  llic  rirrk's  oniop  of  the  DiNtrlcl  Cocirl.  of  tin-  fiiiti-d  str.K.s  for  tlio  KilMitii  Dlstrlcl 

of  I'i'iiii.-ylvaiilii. 


PRE  FACE. 


Tm;  discovery  of  tlio  gront  iaw  of  nature,  the  law  of  ^fravitatioii,  l)y 
Nkwton,  prepared  tlie  way  for  the  hrilliant  aeliieviinents  wliieli  have 
distinguished  the  history  of  astrononii<'al  science.  A  lirst  essential,  how- 
ever, to  the  solution  of  those  recondite  jin>l)h  iii.s  which  were  to  exhihit 
the  etU'ct  of  tlie  nnituul  attraction  of  the  bodies  of  our  sy.-teni,  was  the 
development  of  the  infinitesimal  calculus;  and  the  lal)ors  of  those  who 
devoted  themselves  to  pure  aiuilysis  have  contril)uti'd  a  n\ost  iiiipnriant 
part  in  the  uttninnient  of  the  liiL'h  degree  of  perfection  which  character- 
izes the  resui's  of  astronomical  investigations.  Of  the  earlier  efforts  to 
develop  the  great  results  following  from  the  law  of  gravitation,  <hose  of 
Kli.KK  stand  pre-eminent,  and  the  memoirs  which  he  j)ul)lish(  d  have, 
in  reality,  furnished  the  germ  of  all  sul)se(|uent  investigations  in 
celestial  mechanics.  In  this  eoniiection  al.so  the  names  of  IIkkxoiim.i, 
Ci-AIK.VUT,  and  D'Ai.KMHKUT  deserve  the  most  honorable  mention  as 
having  contributed  also,  in  a  high  degree,  to  give  direetitm  to  tlie  inves- 
tigations which  were  to  unfold  so  many  mysterie.s  of  nature.  By  means 
of  the  researches  thus  inaugurated,  the  great  problems  of  mechanics 
were  suece.«sfully  solved,  many  beautiful  theorems  relating  to  the  planei- 
ury  motions  demonstrated,  and  many  useful  formula'  developed. 

It  is  true,  however,  that  in  the  early  stage  of  the  science  methods 
were  developed  which  have  since  been  found  to  be  impracticalde,  even 
if  not  erroneous;  still,  enough  was  effected  to  direct  attention  in  the 
proper  channel,  and  to  prepare  the  way  for  the  more  oomplett;  labors  of 
La<}U.\N(;e  and  Laplack.  TI.o  genius  and  the  analytical  skill  of  the.«e 
extraordinary  men  gave  to  the  progress  of  Theoretical  Astronomy  the 
most  rapid  strides ;  and  the  intricate  investigations  which  they  suecess- 
fuliy  performed,  served  constantly  to  educe  new  discoveries,  so  that  of 
all  the  problems  relating  to  the  mutual  attraction  of  the  several  jjlanets 

3 


I'HKFACE. 


hut  little  more  roncrmcd  to  be  acconiplislied  by  their  sueeecsors  than  to 
develop  and  .simplify  tlie  iiietliods  which  tiiey  iiiaile  known,  and  to  intro- 
duce sucli  niodiHcations  as  should  be  indicated  l)y  expericnec  or  rendered 
j)ossibh'  by  the  latest  <liscf)veri('s  in  the  domain  of  pure  analysis. 

Th(!  problem  of  <letermiinnf^  the  elements  of  tlie  orbit  of  a  eonict 
moving  in  a  juirabola,  by  mean.s  of  observed  places,  which  had  been 
considered  by  Nkwtox,  KfLKH,  Uuscovicn,  Lamiskht,  and  others, 
received  from  Laohanoe  and  Lai'LAci:  the  most  careful  consideration 
in  the  light  of  all  that  had  been  previously  done.  The  solution  given 
by  the  former  is  analytically  complete,  but  far  from  being  practically 
complete;  that  given  by  the  latter  is  especially  simple  and  practical  so 
fur  as  regards  the  labor  of  computation;  but  the  results  olttained  by  it 
are  so  affected  by  the  unavoidable  errors  of  observation  as  to  be  often 
little  more  than  rude  api)roximations.  The  method  which  was  found  to 
answer  best  in  actual  practice,  was  that  proposed  by  Ol.nKits  in  his 
work  entitled  LclchicMc  vnd  hetjiiemste  Mct/iode  die  Balm  dues  Cometen 
zu  hcnrhnen,  in  which,  by  making  use  of  a  beautiful  theoi-em  of  para- 
bolic motion  demonstrated  by  Ei'LKR  and  also  by  LAMMinn',  and  by 
adopting  a  method  of  trial  and  error  in  the  numerical  .solution  of 
certain  equations,  he  was  enabled  to  ertect  a  solution  which  could  be 
performed  with  remarkable  ease.  The  accuracy  of  the  results  obtaii.od 
by  ()i.!!i:i!s's  method,  and  the  facility  of  its  api)lieation,  directed  th:> 
attention  of  Li:<ii;M)i!i!:,  Ivohy,  Gaiss,  and  KnckI';  to  this  subject,  and 
by  them  the  method  was  extended  an<l  generalized,  and  rentlered  apjdi- 
cable  in  the  exceptional  cases  in  which  the  other  methods  failed. 

It  should  be  observed,  however,  that  the  knowledge  of  one  element, 
the  eccentricity,  greatly  facilitated  the  sidution;  and,  although  elliptic 
elements  had  been  computed  for  some  of  the  comets,  the  lirst  hyi)othesis 
was  that  of  parabolic  motion,  so  that  the  subsequent  process  recpiired 
simply  the  determination  of  the  corrections  to  be  aj)plied  to  these  ele- 
ments in  order  to  satisfy  the  observations.  The  more  difficult  problem 
of  determining  all  the  elements  of  j)lanetary  motion  directly  from  three 
observed  places,  remained  unsolved  until  the  discovery  of  Ceves  by 
PiAzzr  in  1801,  by  which  the  attention  of  CtAI's.s  was  directed  to  this 
subject,  the  result  of  which  was  the  subsequent  publication  of  his 
Theoria  Motus  Corportim  Calcstium,  a  most  able  work,  in  which  he  gave 
to  the  world,  iu  a  finished  form,  the  resulta  of  m  ny  years  of  attention 


lMti;i'A(  E. 


to  tlio  suhjof't  of  wliicli  it  treats.  His  mothnd  for  (Ictcrmining  all  the 
elcnu'iitK  directly  from  given  observed  plaees,  as  jriven  in  the  Tlirorhi 
MoliiK,  and  08  Hnlwe<|nently  ;;iven  in  a  revi.se«l  form  by  Enmri:,  leaves 
scarcely  any  tliinjj  to  be  dc.-ircd  on  this  topic.  In  the  same  work  ho 
gave  the  tirst  explanation  of  the  nietlu)d  of  least  scunires,  u  method 
which  has  been  of  inestinuiblu  service  in  investigutiuns  depending  un 
observed  datu. 

The  discovery  of  the  minor  planets  direi'ted  attention  also  to  the 
methods  of  determininj:;  their  [urturbatioiis,  since  those  applied  in  the 
case  of  till!  major  planets  were  found  to  l)e  inapplicable.  For  a  long 
time  astronomers  were  content  simply  to  com|)nte  the  special  perturba- 
tions of  these  bodies  from  epoch  to  epoch,  and  it  was  not  until  the  com- 
mencomeiit  of  the  brilliant  researches  by  IIanskx  that  serious  hopes 
were  entertained  of  being  abl"  to  compute  successfully  the  general  per- 
turbations of  these  bodies.  I'y  devising  an  entirely  new  mode  of  con- 
sidering the  pertnrltations,  namely,  by  determining  what  may  be  called 
the  perturbations  of  the  time,  and  thus  j)assing  from  the  undisturbed 
place  to  the  disturbed  place,  an<i  by  other  ingenious  analytic:al  and 
mechanical  devices,  he  succeeded  in  effecting  a  solution  of  this  most 
difficult  problem,  and  his  latest  works  contain  all  the  forniuhu  which  aro 
reijuired  for  the  cases  actually  occurring.  The  refined  and  diificult 
analysis  and  the  laborious  calculations  involved  were  such  that,  even 
after  IIansi;n's  methods  were  made  known,  astronomers  still  adhered  to 
the  method  of  special  ])erturbations  by  the  vari.  tion  of  constants  as 
developed  by  Lacihanok. 

The  discovery  of  Ai<(nm  by  IIknckk  was  sj)eedily  followed  by  the 
discovery  of  other  planets,  and  fortunately  indeed  it  so  happened  that 
the  subject  of  special  perturbations  was  to  receive  a  new  improvement. 
The  discovery  by  Bond  and  KNCKt;  of  a  metliod  by  which  we  determine 
at  once  the  variations  of  the  rectangular  co-ordinates  of  the  disturbed 
body  by  integrating  the  fundamental  e(iuations  of  motion  by  means  of 
mechanical  quadrature,  directed  the  attention  of  IIaxskx  to  this  jdiase 
of  the  problem,  and  soon  after  he  gave  I'ormuhe  f  )r  the  determimition 
of  the  perturbations  of  the  latitude,  the  mean  anomaly,  anil  the  loga- 
rithm of  the  radius-vector,  which  are  exceedingly  convenient  in  the 
process  of  integration,  and  which  have  been  found  to  give  the  most 
satisfactory  results.     The  formuhe  for  the  perturbations  of  the  latitude, 


6 


JM{KFA(  K. 


triio  Idiifrltiido,  ami  ra«rni!»-V('ctiir,  to  1»»'  infcfrrati d  in  the  Hnni«'  inunnor, 
wcro  aftcrwanlH  ^.'ivcii  ity  Bui  nxmw. 

Having'  thus  stattfl  hritfly  a  few  lii'torical  fact.'*  rohitin^,'  t(i  thi 
ju'oldciiis  of  theoretical  a.-trniioiiiy,  I  proeeod  to  u  Htatiiiieiit  of  the 
object  of  this  work.  The  discovery  of  m  many  platu't»  ami  comets  has 
furnished  a  wich;  fiehl  for  exercise  in  the  cahMihitions  rehitiii}^  to  their 
motions,  and  it  hnn  occurred  to  me  that  a  work  which  shotihl  contain  a 
development  of  all  the  formul.T  recjuired  in  determininj,'  the  orhitsof  the 
heavenly  hodies  directly  from  given  observed  jjlaces,  and  in  correcting 
these  orbits  by  means  of  more  extended  discussions  of  series  of  observa- 
tions, including  also  the  determination  of  the  jiertu-'bations,  together 
with  a  complete  collection  of  auxiliary  tables,  and  also  such  jiractical 
«lirections  as  might  guide  the  inexj)eriiiiced  computer,  might  add  very 
materially  to  the  progress  of  the  science  by  attracting  the  attention  of  a 
greater  number  of  competent  computers.  Having  carefidly  read  the 
works  of  the  great  imisters,  my  plan  wa.s  to  prepare  a  complete  work  on 
this  .subject,  commencing  with  the  fundamental  principles  of  dynamics, 
and  systematically  treating,  from  one  point  of  view,  all  the  problems 
presented.  The  scope  and  the  arrangement  of  the  work  will  be  best 
understood  after  an  examination  of  it.s  contents;  and  let  it  sufliee  to  add 
that  I  have  endeavored  to  keej)  constantly  in  view  the  wants  of  the 
com])Uter,  providing  for  the  exceptional  ca<es  as  they  occur,  and  giving 
all  the  formulie  which  appeared  to  me  to  be  best  adaj)ted  to  the  problems 
under  consideration.  I  have  not  thought  it  worth  while  to  trace  out  the 
geometrical  signification  of  many  of  the  auxiliary  (juantities  introduced. 
Those  who  are  curious  in  such  matters  may  readily  derive  many  beau- 
tiful theorems  from  a  consideration  of  the  relations  of  some  of  these 
auxiliaries.  For  convenience,  the  formula;  are  numbered  consecutively 
through  each  chapter,  and  the  references  to  those  of  a  preceding  chapter 
are  defined  by  adding  a  subscript  figure  denoting  the  number  of  the 
chapter. 

liesides  having  read  the  works  of  those  who  have  given  special  atten- 
tion to  these  problems,  I  have  consulted  the  Asfronomische  Xachrichfen, 
the  Astronoviical  Jourtin!,  and  other  astronomical  periodicals,  in  which 
is  to  be  found  much  valuable  information  resulting  from  the  experi- 
ence of  those  who  have  been  or  are  now  actively  engaged  in  astro- 
nomical pursuits.     I  must  also  express  my  obligations  to  the  publishers, 


PUKFAIK.  7 

t 

M«'ssiv.  ,1.  H.  Liri'iNcoTT  &  Co.,  for  tli(>  f^oncrous  intorcwt  wliich  tlu'V 
liavc  inaiiil'cstcil  in  tlio  |iiililicatii)ii  ut'  tlio  work,  ami  aUo  to  Dr.  !i.  A. 
Cioii.i),  ol'  ('aiiil)ritlj,'<',  Ma^x.,  aixl  to  Dr.  Ori'oi.zinj,  of  Vifiiiia,  fur 
valiiaMo  sufjfj<'.>*tion.«<. 

For  tlx'  (Ictcnniiiatioii  of  tlic  tiini>  from  tlu>  pcriliclion  aixl  of  tlio  true 
aiioinalv  in  very  ('crcnlrii'  nrhits  I  have  jijivcn  tlio  iiictlio<l  |(ro|)or<«'i|  liy 
lii:ssi:i,  in  tlu"  Mitinitlirhi  ('<iin.yiiiiiil<nz,  v<»l.  xii., — the  taldts  f()r  which 
wcro  ."uljsiMiiK'ntly  ^^ivi-n  by  IJkC.nnow  in  his  AMionomiral  Nullrri, — amt 
also  the  nu'thod  |iro|»oscd  by  (lAl'i^s,  but  in  a  more  convenient  form. 
For  obvious  reason?*,  I  have  ^iven  the  solution  for  the  special  cas(>  of 
paraliolii'  motion  before  eomplefing  the  solution  of  the  general  problem 
of  linding  all  of  the  elements  of  the  orbit  by  means  of  three  observed 
places.  The  ditfercntial  formuhe  and  the  other  formuhe  for  eorreotiiig 
approximate  elements  are  given  in  a  form  convenient  for  appli<ation, 
and  the  formuhe  for  finding  the  chord  or  the  time  of  dcscribi'ig  the 
sul)tende(l  arc  of  the  orbit,  in  the  ea.se  of  very  eccentric  orbits,  will  be 
found  very  convenient  in  practice. 

I  have  given  a  pretty  full  development  of  the  application  of  the 
theory  of  pnjbabilitics  to  the  combination  of  ob.scr  vat  ions,  endeavoring 
to  direct  the  attention  of  the  reader,  as  far  a.s  possible,  to  the  sources  (jf 
error  to  be  apprehen<led  and  to  the  most  advantageous  method  of  treat- 
ing the  problem  so  as  to  elimimite  the  effects  of  these  errors.  For  the 
rejection  of  doubtful  observation.^,  according  to  theoretical  considerations, 
I  have  given  the  sin>ple  formula,  suggested  by  Ciiauvknf.t,  wliich  fol- 
lows directly  from  the  fundamental  equations  for  the  probability  of 
errors,  and  which  will  answer  for  the  purposes  here  required  as  well  a^ 
the  more  complete  criterion  proposed  by  Pr.iitcK.  In  the  chapter 
devoted  to  the  theory  of  special  jierturbations  I  have  taken  particular 
pains  to  develop  the  whole  subject  in  a  coni[)lete  and  practical  form, 
keeping  constantly  in  view  the  requirements  for  accurate  and  convenient 
numerical  application.  The  time  is  adopted  as  the  independent  variable 
in  the  determination  of  the  perturbations  of  the  elements  directly,  .since 
experience  has  established  the  convenience  of  this  form;  and  should  it 
be  desired  to  change  the  independent  variable  and  to  use  the  diiTerential 
coefficients  with  respect  to  the  eccentric  anomaly,  the  equations  betweeit 
this  function  and  the  mean  motion  will  enable  us  to  effect  readily  the 
required  transformation. 


4 


rilKKAcK. 


Till'  nuiiinical  cxniiiph's  invulvc  ilalii  (Urivi)!  iVuin  actual  oliscrva- 
tioiis,  and  care  lias  liocii  takrii  to  iiinkr  tlicni  coiiiiiK  tc  in  every  rc.<<|K<ct, 
HO  nx  to  svr\e  m  a,  f^u'ulo  to  tlio  oHiirts  of  tlionc  not  familiar  with  tlicn* 
calculaliouri;  and  wlion  ditrcri-nt  t'lindanictital  planex  arc  npokcii  ol',  it  ii^ 
])r«->iiiiu>d  that  tliu  rcadiT  IH  familiar  with  the  eltniciitrt  of  Hphcrii-al 
aHtroiiomy,  xi  that  it  is  nnnccosmiry  to  Htutc,  in  all  ca-xos,  whcthor  tiio 
contro  of  the  nphorc  is  taken  at  the  centre  of  the  earth,  or  at  any  other 
point  in  space. 

The  preparation  of  the  Tahlcs  has  cost  mo  n  jrront  nmonnt  of  labor, 
logarithms  of  ten  decimals  heinj:  employed  in  order  to  he  sure  of  the 
last  decimal  {.dven.  Several  of  those  in  previous  use  luvvt!  been  recom- 
puted ami  extended,  and  others  here  ;.'iveii  for  the  lirst  time  liave  been 
prepared  w  ith  spec  iai  care.  The  adopted  value  of  the  constant  of  the 
solar  attraction  is  that  j;iven  by  ('Aiss,  which,  as  will  appear,  is  not 
accurately  in  accordance  with  the  adoption  of  the  mean  distance  of  the 
earth  from  the  sun  as  the  unit  cf  spncu;  but  until  the  absolute  value  of 
the  earth's  mean  motic  n  is  known,  it  is  best,  for  the  sake  of  uniformity 
and  accuracy,  to  retain  (JAiS't's  constant. 

The  j)riparation  of  this  wcrk  has  been  eflected  amid  many  interrup- 
tions, and  with  other  labors  constantly  j»rcssin^'  mc,  by  wliii-h  the  itroj,M*ess 
of  its  ])ublication  has  been  somewhat  delayed,  even  since  the  stereo- 
tyj)ing  was  commenced,  so  that  in  some  eases  I  have  been  anticipated 
in  the  publication  of  formuhe  which  wouhl  have  here  appeared  for  the 
first  time.  I  have,  however,  endeavored  to  perform  conscientiously  the 
self-imposed  task,  seeking  always  to  secure  a  logical  sequence  in  the  de- 
veloj)ment  of  the  formuhe,  to  preserve  unitbrmity  and  elegance  in  the 
notation,  and  to  elucidate  the  successive  steps  in  the  analysis,  so  that  the 
work  may  be  read  by  those  who,  possessing  a  respectable  mathematical 
education,  desire  to  be  informed  of  the  means  by  whicli  astronomers  are 
enabled  to  arrive  at  so  many  grand  results  connected  with  the  motions 
of  the  heavenly  bodies,  and  by  which  the  grandeur  and  sublimity  of 
creation  arc  unveiled.  The  labor  of  the  i>re])aration  of  the  work  will 
have  been  fully  repaid  if  it  shall  be  the  menus  of  directing  a  more 
general  attention  to  the  study  of  tlie  wonderful  mechanism  of  the  hea- 
vens, the  contemplation  of  which  must  ever  serve  to  imi)ress  ujjon  the 
mind  ihc  reality  of  the  perfection  of  the  omnh'otent,  the  living  GOD! 

Observatouy,  Ann  Aubob,  June,  1867. 


CONTENTS. 


TIIEOllETIOAL  ASTRONOMY. 


CIIAPTKIl  I. 


» 


INVKsTKi.VTt(V  V  TMK  ITNDAMr.NTAI,  K<Jf.\TI0N9  OF  MOTION,  ASM)  OK  Till:  lou- 
Ml  I,.K  Killl  IiKTi:ilMIMN(i,  KUuM  KXuWX  DI.KMIATS,  TlIK  111:1. liK  KNIKIC  AM) 
(iKurKNTIlIC  I'LA(  ra  OF  A  II KA  VKNI.V  IIOIlY,  AOAI-IKK  To  JJl  MKItK  AL  (  OM- 
ri'TATION    KOK  CASKS  OF   ANY    KCCKXTIULITY    WlIATKVKIt. 

rAiif. 

FiiiidaiiK'HtMl   I'riiu'ipK''^ 1-') 

Attr;ii'ti(»ii  »(  Splu-ron Ht 

M(iti(iii.-«  <i(  II  Systi'in  of  Jindiis *j;{ 

liiVMiiiiMc  I'liiiu'  (if  tlif  Systfin 'Jit 

Motini)  of  ;i  Siiliil  I>o(ly .".l 

Till'  I'liits  of  Spaci',  TiiiU',  ;iii(l  Mass ;i(i 

Motion  of  a  liody  ivlativc  to  the  Sun :',X 

Kijiiations  for  ('iiilixtiirhcd  Motion  -I'i 

iK'tiTiniiiation  of  tlio  Attractivo  Vorco  of  tiu-  Sun  4!t 

Dotcrniinaiion  of  tlio  IMace  in  an  I'^lliptic  Oi'i)it ')',i 

Dctiiinination  of  liic  I'luct  in  a  I'araliolic  OiMt oO 

lU'ti'i-ininiitiiPii  of  llic  Piaco  in  a  iry]n'i'tiolic  ()ri)it ti') 

Mctiiodn  I'or  tindiiif;  tlie  True  Anomaly  and  tlicTiine  from  tiie  IVriiiuiion  in  tiie 

case  of  <  trliits  of  (rivat  Kcccntricity 70 

Di'ti-rniination  of  tlie  I'oHition  in  Spaii' SI 

IK'lioiviitiii.'  Longitude  and  Latitude S3 

Ri'dnction  to  tlio  Kdiptic 85 

Cii'ooentrif  Li)iif{itii(le  and  Latitude Sfi 

Transformation  of  Spherical  Co-ordinates 87 

Direct  Determination  of  the  Geocentric  Kiglit  Ascension  and  Declination 90 

Reduction  of  the  Element*  from  one  Kpoch  to  another 09 

Numerical  Kxamples 103 

InterpoUtioii  ^ ll'J 

Time  of  Oiiposition 114 


10 


CONTENTS. 


CHAPTER  II. 

INVKSTIOATION  OF  TIIK  DJFFERKNTIAL  FORMt:i,-K  WltlCII  EXPRESS  THE  KELATIOJf 
HETWEEN  THE  C.EOC'EXTRIC  OR  IIKI KKENTRIC  PI-ACJK  OF  A  IIEAVENEY  BODY 
AND  THE  VARIATIONS  OF  THE  ELEMENTS  OF  ITS  OKBIT. 

PAGE 

Variatii)ii  of  tlic  Right  Ascension  and  Declination 118 

Case  of  Piirabolic  Motion  125 

Case  of  Hyperhoiic  Motion 128 

Case  of  Oil.its  didorinj,'  l)iit  little  from  the  Parabola 1.30 

Tsuiiicrical  Examples 135 

Variation  of  the  J^onKitude  an<l  Latitiiile 143 

The  I'lU'ini'nt^  referred  to  the  same  Fundamental  Plane  a^4  the  f  leoceiitrie  Places  H9 

Numerical  Example 150 

Plane  of  the  Orbit  taken  as  the  Fundamental  Plane  to  which  the  Geocentric 

Places  are  referred  153 

Numerical  Example 159 

Vai-iation  of  the  Auxiliaries  for  the  Eijuator 1C3 


Fun<lam 

Formuin 

Modifica 

Determi 

Case  of 

Position 

Fornud 

Formida 

Final (^ 

Determi  1 

Numeric 

Correct  io 

Ai>iiroxii 


CHAPTER  III. 

INVESTIGATION  OF  FOP.MVIwK  FOR  COMPl'TINO  THE  ORBIT  OF  A  COMET  MOVING 
IN  A  PARABOLA,  AND  FOR  COUUIX'TINa  APPROXIMATE  ELEMENTS  BY  THE 
VARIATION   OF  THE    JEOCENTRIC   DISTANCE. 

Correcti(i.i  of  the  Observations  for  Parallax 167 

Fundamental  Equations 169 

Particular  Cases 172 

Ratio  of  Two  Curtate  I)istanees 178 

Determination  of  the  Curtate  Distances 181 

Relation  between  Two  Radii-Vectorcs,  the  Chord  joining  their  Extremities,  and 

the  Time  of  describing  the  Parabolic  Arc 184 

Determination  of  the  Node  and  Inclination 192 

Perihelion  Distance 'uid  Longitude  of  the  Perihelion 194 

Time  of  Perihelion  Passage 195 

Numerical  Example 199 

Correction  of  Approximate  Elements  by  varying  the  Geocentric  Distance 208 

Numerical  Example 213 


CHAPTER  IV. 


DETERMINATION,  FROM  THREE  COMPLETE  OBSERVATIONS,  OF  THE  ELEMENTS  OF 
THE  ORBIT  OF  A  HEAVENLY  BODY,  INCLUDING  THE  ECCENTRICITY  OB  FORM 
OF  THE  CONIC   SECTION. 

Reduction  of  the  D.ita 220 

CorrectioiiH  for  Parallax 223 


CONTEXTS.  11 

PAGE 

Fimdampntal  Eiiuations 225 

I'onnulii'  for  tlio  Curtate  Distaiurs 228 

Modification  of  tiie  Forimiiu'  in  Particular  Cases 231 

lU'teriiiinatioii  of  the  Curtate  Distance  for  tlie  Middle  ()l)servati()n 2.'!() 

Case  of  a  l)oul)le  Solution 2;5!} 

Position  indicated  liy  tlie  Curvature  of  the  Observed  Path  of  the  llody 212 

Fornuilfe  for  a  Second  A)i]iroxirnati')n 243 

Foniudie  for  lindiiijf  the  Hatio  of  the  Sector  to  the  Trianjjle 247 

Final  Correction  for  Aherration 2o7 

Determination  of  the  Flenients  of  the  Orliit 2V,) 

Numerical  Example 204 

Correction  of  the  First  llyjiothesis 278 

Approxinuite  Method  of  finding  the  Itatio  of  the  Sector  to  the  Triangle 271) 


CHAPTER  V. 

DETrnjnxATiox  of  the  onniT  of  a   he.vvenia'  body  from  fofu  obsf-rva- 

TIUXS,   OF   WHICH  THE  SECOND   AND  THIRD  MIST    BE  COMPLETE. 

Fundamental  F(|uations 282 

Deiermination  of  the  Curtate  Distances 28!) 

Successive  Approximations 2!)3 

Determination  of  the  J^lementu  of  the  Urhit 21)4 

Numerical  ICxaiuple 294 

Method  for  the  Final  Approximation 307 


CHAPTER  VI. 

INVESTIGATION   OF   VARIOI'S   FORMFL.F    FOR    THE    CORRECTION    OF    TIIE   APPROXI- 
MATE  ELEMENTS   OF   THE   ORHIT  OF   A    HEAVENLY   BODY. 


Determination  of  the  Elements  of  a  Circular  ()rl)it 

Variation  of  Two  (leoccntric  Distances 

DiHerential  Formula' 

Plane  of  the  Orbit  taken  as  the  Fundamental  Plane 

Variation  of  the  Node  and  Inclination 

Variation  of  One  Geocentric  Distance 

Determination  of  the  Elenieiii3  of  'he  Orbit  by  means  of  the  Co-ordinates  and 

Velocities  

Correction  of  the  Ephemeris 

Final  Correction  of  the  Elements 

Relation  between  Two  Places  in  the  Orbit 

Modification  when  ihe  Semi-Tranavcrse  Axis  is  very  large 

Modification  for  Hyperbolic  Motion 

Variation  of  the  Semi-Transverse  Axis  and  Ratio  of  Two  Curtate  Distances 


311 
313 
318 
320 
324 
328 

332 
33.-, 
338 
339 
341 
34r, 
349 


12  CONTEXTS. 

PAOR 

Variation  of  tlie  Geocentric  Distance  aiid  of  the  Reciprocal  of  the  Senil-Trans- 

vcrse  Axis ;5')2 

K(|nationH  of  Comlitioii IJijS 

{)rl)it  of  ii  Comet..  .'J55 

Variation  of  Two  Kudii-Vectoreri 357 


CHAPTER  VII. 

METHOD  OF  LEAST  SQUARES,  THEORY  OP  THE  COMBINATION  OF  OBSERVATIONS, 
AND  DETERMINATION  OF  THE  MOST  PROBABLK  SYSTEM  OF  ELEMENTS  I'UOM 
A   SERIES   OF   OBSERVATIONS. 

Statement  of  tiie  Problem .^fiO 

Fundiiniental  Ivjuations  for  the  Proliahility  of  Error,* .Sli'J 

Determination  of  the  Form  of  the  Function  which  expresses  the  Probability  ...  303 

The  Measure  of  Precision,  and  the  Probable  Error StitJ 

Distriluition  of  the  Errors 3(>7 

The  Mean  Error,  and  tlie  Mean  of  the  Errors o(!8 

The  Probable  Error  of  the  Arithmetical  Mean .370 

I)eterniiiiation  of  the  Mean  and  Probable  Errors  of  t)bservations 371 

Wei>.dits  of  ()i)served  Values 372 

E(| nations  of  Condition 376 

Isoniial  Iviuations ,378 

Metiiod  of  Elimination , 380 

Detcrminaticm  of  the  Weights  of  the  Resulting  Values  of  the  Unknown  Quanti- 
ties   38R 

Separate  Determination  of  the  Unknown  Quantities  and  of  their  Weights 30'J 

Relation  between  the  Weiglits  and  the  Determinants 396 

Casein  which  the  Problem  is  nearly  Indeterminate 398 

!Mean  and  Probable  Errors  of  the  Results 399 

Cond)i nation  of  C)l)servations 401 

Errors  peculiar  to  certain  Observations 408 

Rejection  of  ])oid)tful  Observations 410 

Correction  of  the  Elements 412 

Arrangement  of  the  Xtnnerical  Operations 4Io 

Kumerical  Example 418 

Case  of  very  Eccentric  Orbits 423 


CHAPTER  VIII. 

INVESTIGATION  OF  VARIOUS  FORMULAE  FOR  THE  DETERMINATION  OF  TIIE  SPECIAL 
rERTURBATION.S   OF   A   HEAVENLY   BODY. 

Fundamental  Equations 426 

Statement  of  the  Problem 428 

Variation  of  Co-ordinates 429 


CONTEXTS.  13 

PAQE 

Mcphaiiiral  Quadrature 431? 

Tlie  Interval  fur  (^nadratiiri' 44.'$ 

Mode  of  c'lli'c-tiiif?  tin-  Inti'^rration 44') 

lVrtiirl)ations  dc-]H>iuling  on  tlie  S<iuari's  and  Higher  Powers  of  tlie  Masses 440 

Numerical  Kxatiii)le 44S 

CiiaiiKe  (if  the  Kiiiiinox  and  Kcliptic 4')5 

iMerniination  of  New  Osenhiting  Klenients 4"i!t 

Variation  of  Polar  Co-ordinates 4(12 

Determination  of  the  Components  of  the  I)istiirl)ini,'  Force 4<)7 

JX'termination  of  the  Heliocentric  or  {jeocontrie  Place 471 

Xnmerical  Kxaniple 474 

Change  of  the  Osculating  Elements 477 

Variation  of  the  Mean  Anomaly,  the  Kadius-Vuctor,  and  the  Co-ordinate  z 4S0 

Fundamental  Ktjuations 4.S:{ 

Determination  of  the  Comjionents  of  the  Disturliing  I'orce 4S!t 

Case  of  very  Kccenlric  <  )r!)its 4!i;{ 

Determination  of  the  Place  of  the  Disturbed  Pody Ill't 

Variation  of  the  Node  and  Inclination .")()2 

Numerical  Example 'A)^) 

Change  of  the  Osculating  Elements old 

Variation  of  Constants olti 

Case  of  very  Eccentric  Orliits .V2;> 

Variation  of  the  Periodic  Time .VJtj 

Numerical  Example o'i!) 

Forinnhe  to  he  used  when  the  Eccentricity  or  the  Inclination  is  small '>',V,i 

Correction  of  the  Assumed  Value  of  the  Disturbing  Ma;<s .");{") 

Perturbations  of  Comets Ty'M 

Motion  about  the  Common  Centre  of  (iravity  of  the  Sun  and  I'lanet ."),'i7 

Reduction  of  the  Fleinents  to  the  Common  Centre  of  (iravity  of  the  Sun  an<l 

Planet  538 

Reduction  by  means  of  Dillerential  Fornuda- !i40 

Near  Approach  of  a  Ccunet  to  a  Planet 54(1 

The  Sun  may  be  regarded  as  the  Disturbing  I'ody 548 

Determination  of  the  Elements  of  the  Orbit  about  the  Planet 550 

>Subse([uent  Motion  of  the  Comet 551 

Efleet  of  a  Resisting  Medium  in  Space 552 

Variation  of  the  Elements  on  accoinit  of  the  Resisting  Medium 554 

Method  to  be  applied  when  no  Assumption  is  made  in  regard  to  the  Density  of 

the  Ether 55fi 


14 


CONTENTS. 


TABLES. 


9 


PAflE 

T.  Angle  of  the  Vertical  and  Logarithm  of  tlie  Earth's  Radiiia oGl 

II.  For  converting  Intervals  of  Mean  Solar  Time  into  Kquivalent  Intervals 

of  Sidereal  Time 5('k5 

III.  For  coiiviTting  Intervals  of  Sidereal  Time  into  JMinivalent  Intervals 

of  Mean  Solar  Time 564 

IV.  For  converting  Hours,  Miiuites,  and  Seconds  into  Decimals  of  a  Day...  505 
V.   For  finding  the  Number  of  Days  from  the  Beginning  of  the  Year 565 

VI.  For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a 

ParaholicOrhit 566 

VII.  For  finding  the  True  .\nomaly  in  a  I'araholic  <)ri)it  when  r  is  nearly  1S0°  til  1 

VII 1.   For  finding  the  Time  from  the  Perihelion  in  a  Paraliolic  Oi-hit C12 

IX.   For  tinding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  Orhits 

of  <  Jreat  Ivccentricity 614 

X.   For  finding  the  True  Anonuily  <>t  the  Time  from  the  Perihelion  in  El- 
liptic and  IIyi)erholic  Orhits 618 

XI.  For  the  Motion  in  a  Paraholic  Orhit (ill) 

XII.   For  the  Limits  of  the  Koots  of  tlie  lC(ination  sin  (2' —  C)  ■-  JHq  f*''!*-'"  f'-- 

XHl.   For  finding  the  Ratio  of  the  Sector  to  the  Triangle 6"J4 

XIV.   For  finding  the  Ratio  of  tlie  Sector  to  the  Triangle G'lO 

X\'.   For  Ell ii>tic  Orhits  of  (ireat  Eccentricity (V.Vl 

XVI.   For  Ilyiierholic  Orhits (1:52 

XVIL   For  Sjiecial  Pert urhat ions (V.'m 

XVIIl.   Elements  of  the  Orhits  of  the  Comets  which  have  heen  ohfeerved 6;?8 

XIX.   Elements  of  the  Orhits  of  the  Minor  Planets 616 

XX.  Elements  of  the  Orhits  of  the  Major  Planet« 618 

XXL  Constants,  &c 649 

Explanation  of  the  Table-s G51 

ArriiNUix. — Precession 657 

Nutation 658 

Aberration  65!) 

Intensity  of  Light 660 

Numerical  Calculations 662 


THEORETICAL  ASTRONOMY. 


CHAPTER  I. 


IXVK«TIfiATION  OF  THE  FUNDAMKN'TAL  KljrATIOXS  OF  MOTION',  AND  OF  TIIK  I-Oll- 
Mll.-K  Kiill  DKTKKMIMXli,  I'lJoM  KNltWN  KLKMKNTS,  I'lIK  lIKMiK  KNTlilc  AM) 
(!K(R'KNTltIf  ri-ACKS  OF  A  IIKAVKM.V  IIODY,  AUAll'KU  TO  NLMKUUAI-  CDMl'lTA- 
TIOK  FOR  (.ASKS  OF  ANY  KCCEXTllK  ITV  WHA TKVKU. 

1.  Tin;  Study  of  the  motion.s  of  the  heavenly  hotlics  dues  not  re- 
quire that  we  should  know  the  ultiniate  limit  of  divisibility  oi'  the 
matter  of  which  they  are  composed, — whether  it  may  \h'  sulxlivided 
indefinitely,  or  whether  the  limit  is  an  indivisible,  im|)enetral)le  atom. 
Nor  are  we  concerned  with  tlic  relations  which  exist  between  the 
separate  atoms  or  molecules,  except  so  far  as  they  ibrm,  in  tiie  a«fgre- 
gate,  a  definite  body  whose  relation  to  other  l)odies  of  the  system  it 
is  required  to  investigate.  On  the  contrary,  in  considering  the  ope- 
ration ol"  ihe  laws  in  obedience  to  which  matter  is  aggregated  into 
single  bodies  and  systems  of  bodies,  it  is  sufKcient  to  conceive  simply 
of  its  divisi!  '"W  to  a  limit  which  may  be  regarded  as  infinitesimal 
compared  with  the  finite  volume  of  the  body,  and  to  regard  the  mag- 
nitude ol'  the  element  of  matter  thus  arrived  at  as  a  matlKunatieid 
point. 

An  element  of  matter,  or  a  material  body,  cannot  give  itself 
motion;  neither  can  it  alter,  in  any  manner  whatever,  any  motion 
which  may  have  been  connnunicated  to  it.  This  tendency  of  matter 
to  resist  all  changes  of  its  existing  state  of  rest  or  motion  is  known 
as  incrtid,  and  is  the  fundamental  law  of  the  motion  of  bodies.  Iv\- 
perience  l:ivariably  confirms  it  as  a  law  of  nature;  the  continuance  of 
motion  as  resist^mces  are  removed,  as  well  as  the  sensibly  unchanged 
motion  of  the  heavenly  bodies  during  many  centuries,  affording  the 

U 


16 


THEORETICAI.   A8TK()X0>[Y. 


*| 


m 


'HI 


iii|i 


most  convinciii};  pioof  of  it.«i  iiiiivortiality.  Whoiievcr,  tlicreforo,  a 
ujutcrial  point  cxpc'ricnccs  any  clianfjje  ol'  its  state  as  respects  rest  or 
motion,  the  eause  must  l)e  attributed  to  the  operation  of  something 
external  to  tlie  element  itself,  and  which  we  desij^nate  by  the  word 
force.  The  nature  of  forces  is.  ffcnerally  unknown,  and  wt;  estimate 
them  by  the  efl'ects  which  they  produce.  They  are  tluis  rendered  com- 
parable with  some  unit,  and  may  be  expressed  by  abstract  numbers. 

2.  If  a  material  point,  free  to  move,  receives  an  impulse  by  virtue 
of  the  action  of  any  force,  or  if,  at  any  instant,  the  force  by  which 
motion  is  conununicated  shall  cease  to  act,  the  subsequent  motion  of 
the  point,  according  to  the  law  (tf  inertia,  nmst  be  rectilinear  and 
nnlfonn,  equal  spaces  being  described  in  equal  times.  Thus,  if  s,  r, 
and  t  represent,  resi)ectivcly,  the  ^jtace,  the  vclodty,  and  the  Iliac,  the 
measure  oi'  v  being  the  space  described  in  a  unit  of  time,  we  shall 

have,  in  this  case, 

8  =  vt. 

It  is  evident,  however,  that  the  space  described  in  a  unit  of  time  will 
vary  with  the  intensity  of  the  force  to  which  the  motion  is  tlue,  and, 
the  nature  of  the  force  being  unknown,  we  must  necessarily  compare 
the  velocities  (lommunicated  to  the  point  by  ditf'erent  I'orces,  in  order 
to  arrive  at  the  relation  of  their  elT'ects.  We  are  thus  led  to  regard 
the  ibrce  as  proportional  to  the  velocity;  and  this  also  has  received 
the  most  indubitable  proof  as  being  a  law  of  nature.  Hence,  the 
principles  of  the  composition  and  resolution  of  forces  may  be  applied 
also  to  the  composition  and  resolution  of  velocities. 

If  the  force  acts  incessantly,  the  velocity  will  be  accelerated,  and 
the  force  which  produces  this  motion  is  called  an  accclenitinf/  force. 
In  regard  to  the  mode  of  operation  of  the  force,  however,  we  may 
consider  it  as  acting  absolutely  witijout  cessation,  or  we  may  regard 
it  as  acting  instantaneously  at  successive  infinitesimal  intervals  repre- 
sented by  dt,  and  hence  the  motion  as  uniform  during  each  of  these 
intervals.  The  latter  supposition  is  that  which  is  best  adapted  to 
the  requirements  of  the  intinitesimal  calculus;  and,  according  to  the 
fundamental  principles  of  this  calculus,  the  finite  result  will  be  the 
same  as  in  the  case  of  a  force  whose  action  is  absolutely  incessant. 
Therefore,  if  we  represent  'the  element  of  space  by  (Is,  and  the  ele- 
ment of  time  by  dt,  the  instantaneous  velocity  will  be 

ds 


"i 


which  will  vary  from  one  instant  to  another. 


FUNDAM KXTA  L    IMM XCI IM-RS. 


17 


may 


3.  Siiirc  tlio  force  is  proportional  to  the  velocity,  its  nu'asurc  at 
aiiv  instant  will  ho  tU'tcrniiiicd  l)y  the  correspond inj;  velocity.  It' 
the  accelerating  force  i.s  constant,  the  motion  will  he  nniformly  accele- 
rated; and  if  we  designati'  the  acceleration  due  to  the  force  by/,  the 
unit  of/ being  the  velocity  generated  in  a  unit  of  time,  we  shall  have 

V  =:ft. 

If,  however,  the  force  be  variable,  we  shall  have,  at  any  instant, 
the  relation 

J-  dt' 

the  force  being  regarded  as  constant  in  its  action  during  the  elemeut 
of  time  (It.     The  instantaneous  value  of  v  gives,  by  ditferentiation, 


and  hence  we  derive 


dv 

'dt~ 

(/'.« 

w 

/= 

d'.i^ 

df' 

CD 


so  that,  in  varied  motion,  the  acceleration  due  to  the  force  is  mea- 
sured by  the  second  ditt'erential  of  the  space  divided  by  the  sc^uare 
of  the  element  of  time. 

4.  By  the  tiumi  of  the  body  we  mean  its  absolute  quantity  of  mat- 
ter. The  den.slti/  is  the  mass  of  a  unit  of  volume,  and  hence  the 
entire  niass  is  equal  to  the  volume  multiplied  by  the  density.  If  it 
is  required  to  compare  the  ibrces  which  act  upon  ditfercnt  bodies,  it 
is  evident  that  the  masses  must  be  considered.  W  equal  masses 
receive  impulses  by  the  action  of  instantaneous  forces,  the  forces 
acting  on  each  will  be  to  each  other  as  the  velocities  imparted  ;  and 
if  we  consider  as  the  unit  of  Ibrce  that  which  gives  to  a  unit  of  mass 
the  unit  of  velocity,  we  have  f)r  the  measure  of  a  force  F,  denoting 
the  mass  bv  31, 

F^Mv. 

This  is  called  the  quanfitj/  of  motion  of  the  body,  and  expresses  its 
capacity  to  overcome  inertia.  By  virtue  of  the  inert  state  of  matter, 
there  can  be  no  action  of  a  force  without  an  equal  and  contrary  re- 
action; for.  if  the  body  to  which  the  force  is  applied  is  fixed,  the 
equilibrium  between  the  resistance  and  the  force  necessarily  implies 
the  development  of  an  equal  and  contrary  force;  and,  if  the  body  be 
free  to  move,  in  the  change  of  state,  its  inertia  will  ojjpose  c-cpial  and 

2 


1» 


TIIEOHKTICAL    ASTKOXOMY. 


:l      ! 


oontniry  ri'si.shuu'o.     Ilencc,  as  iv  lu'ccssaiy  ('(tnso(|ii(Mi('o  of  inertia,  it 

follows  that  action  and  ivactiori  aio  siinnltancous,  e<)nal,  and  contrary. 

If  the  body  is  acted  upon  hy  a  force  su(!li  that  tlie  motion  is  varied, 

the  aeceleratinjjj  forct;  upon  eacli  element  t)f  its  mass  is  represented  by 


do 
dt 


,  and  tiie  entire  mothr  force  F  is  expressed  by 

dv 


F=M 


dt' 


M  being  the  sum  of  all  the  elejuents,  or  the  mass  of  the  body.   Since 

ds 

v  = 

this  gives 


dt' 


which  is  the  expression  for  the  intensity  of  the  motive  force,  or  of 
the  force  of  inertia  developed.  For  the  unit  of  mass,  the  measure 
of  the  force  is 

dt:' ' 

and  this,  therefore,  expresses  that  part  of  the  intensity  of  the  motive 
force  wliich  is  impressed  upon  the  unit  of  mass,  aud  is  what  is  usually 
called  the  accdc rating  force, 

5.  The  force  in  obedience  to  which  the  heavenly  bodies  perform 
their  journey  through  space,  is  known  as  the  attraction  of  f/rucitation; 
and  the  law  of  the  operation  of  this  i()rce,  in  it^self  simple  and  unique, 
has  b(!en  confirmed  and  generalized  by  the  accumulated  researches  of 
modern  science.  Not  only  do  we  find  that  it  controls  the  motions  of 
the  bodies  of  our  own  solar  system,  but  that  the  i*evolutions  of  binary 
systems  of  stars  in  the  remotest  regions  of  space  proclaim  the  uni- 
versality of  its  operation.  It  unfailingly  exi)lains  all  the  phenomena 
observed,  and,  outstripi)ing  observation,  it  has  furnished  the  means 
of  predicting  many  phenomena  subsequently  observed.  The  law  of 
this  force  is  that  everi/  particle  of  matter  i,s  attracted  by  every  other 
particle  by  a  force  which  varies  directly  as  the  iiias8  and  inversely  as 
the  square  of  the  distance  of  the  attracting  particle. 

This  reciprocal  action  is  instantaneous,  and  is  not  modified,  in  any 
degree,  by  the  interposition  of  other  particles  or  bodies  of  matter.  It 
is  also  absolutely  independent  of  the  nature  of  the  molecules  them- 
selves, and  of  their  aggregation. 


Arrii.vcTiox  op  si'iiereh. 


19 


If  wo  consider  two  bodies  the  mnsscs  of  wliich  arc  hi  and  in',  and 
wliosc  inagnitudts  are  .so  small,  relativt^ly  to  their  mutual  distance  y, 
that  we  may  ri'jj;ard  them  as  material  points,  aceordinjr  to  the  law  of 
gravitation,  the  aetion  of  in  on  each  molecule  or  unit  of  m'  will  bo 


in 


and  the  totiil  force  on  ni'  will  bo 


% 


Hi 


,m 


m 


The  ai'tion  of  m'  on  each  molecule  of  in  will  bo  expressed  by  — -,  and 
its  total  action  by 


m 


m 


The  absolute  or  movinj?  force  with  which  tho  masses  in  and  m'  tend 
toward  each  other  is,  therefore,  the  same  on  each  body,  which  result 
is  a  necessary  consequence  of  the  ('(juality  of  action  and  reaction. 
The  velocities,  however,  with  which  these  bodies  would  approach 
each  other  must  be  different,  the  velocity  of  the  smaller  mass  exceed- 
ing; that  of  the  greater,  and  in  the  ratio  of  the  masses  moved.  The 
expression  for  the  velocity  of  la',  which  would  be  generated  in  a  unit 
of  time  if  the  force  remained  constant,  is  obtained  by  dividing  the 
absolute  force  exerted  by  in  by  the  mass  moved,  which  gives 

m 

7 

and  this  is,  therefore,  the  measure  of  the  acceleration  due  to  the 
action  of  in  at  the  distance  f».  For  the  acceleration  due  to  the 
action  of  tn'  we  derive,  in  a  similar  manner, 

6.  Observation  shows  that  the  heavenly  bodies  arc  nearly  spherical 
in  form,  and  we  shall  therefore,  preparatory  to  finding  the  equations 
wliich  express  the  relative  motions  of  the  bodies  of  the  system,  de- 
termine the  attraction  of  a  spherical  mass  of  uniform  density,  or 
vaiying  from  the  centre  to  the  surface  according  to  any  law,  for  a 
point  exterior  to  it. 

If  we  suppose  a  straight  line  to  be  drawn  through  the  centre  of  the 
sphere  and  the  point  attracted,  the  total  action  of  the  sphere  on  the 
point  will  be  a  force  acting  along  this  line,  since  the  mass  of  the 
sphere  is  symmetrical  with  I'espect  to  it.     Let  dm  denote  an  element 


20 


TH  KOIIETICA  L   AHTHONOM Y. 


'i.i;n 


of  the  inushof  the  sphere,  ami  (>  its  dibtunce  from  the  pouit  attracted} 

then  will 

dm 

expres.s  the  ivetioii  of  this  element  on  the  point  uttrm*te<l.  If  we  sup- 
pose tile  density  of  tlie  sphere  to  be  constant,  and  c([ual  to  unity,  the 
clement  dia  Ijceomes  an  element  of  volume,  and  will  be  expressec'  oy 

dm  =  dx  dij  dz ; 

X,  y,  and  z  being  the  co-ordinates  of  the  clement  referred  to  a  system 
of  rectangular  co-ordinates.  If  we  take  the  origin  of  co-ordinates 
at  the  centre  of  the  sphere,  and  introduce  polar  co-ordinates,  so  that 

X  =  r  cos  tp  cos  0, 
y  :::=r  COS  (f  siu  0, 
z  :^r  sin  ^, 

the  expression  for  dm  becomes 

dm  =  }•*  cos  f  dr  d<p  dO ; 
and  its  action  on  the  point  attracted  is 

,.      r"  cos  V  dr  dv  dO 
"/= 1 

If  we  suppose  the  axis  of  z  to  be  directed  to  the  point  attracted, 
the  co-ordinates  of  this  point  will  be 


0, 


2^  =  0, 


2  =  a. 


a  being  the  distiuicc  of  the  point  from  the  centre  of  the  sphere,  and, 
since 

p'-={x-  xj  +  (2/  -  ]fy  +  (3  -  z'y, 

we  shall  have 

p*  =  a'^  —  2ar  sin  <p  -\-  r^. 

The  component  of  the  force  dj  in  the  direction  of  the  line  a,  join- 
ing the  point  attracted  and  the  centre  of  the  sphere,  is 

dfcosr, 

where  y  is  the  angle  at  the  point  attracted  between  the  element  dm 
and  the  centre  of  the  sphere.  It  is  evident  that  the  sum  of  all  the 
components  which  act  in  the  direction  of  the  line  a  will  express  the 
total  action  of  the  sphere,  since  the  sum  of  those  which  act  perpeu- 


ATTRACTION   OF   SPHEnES. 


91 


(linilar  to  this  line,  taken  so  as  to  include  the  entire  mass  of  the 

sphere,  in  zero. 

IJut  we  have 

a  =  z  -j-  p  cos  Y, 
and  hence 


008^  = 


a  —  rsni^ 


The  differentiation  of  the  expression  for  (i^,  with  respect  to  a,  gives 
dp       a  —  rsiny 


da 


cos  y. 


Therefore,  if  we  denote  the  attraction  of  the  sphere  by  A,  we  .shall 
have,  by  means  of  the  values  of  ((/'and  cos^, 


or 


.  .  r*  cos  <p  df  df  do    dp 

p'  da 


dl 

dA  =  —  r'  cos  V  dr  d<p  dO  7-. 

da 


The  polar  co-ordinates  r,  <f,  and  d  are  independent  of  a,  and  hence 

,  r'  c'0S(;>  clr  dif>  dff 


Let  us  now  put 
and  we  shall  have 


dA^ 


dV-. 


da 
r*  cos  <p  dr  d<p  dO 


A  =  - 


P 

dV 

da 


(2) 


Consequently,  to  find  the  total  action  of  the  sphere  on  the  given 
point,  we  have  only  to  find  V  by  means  of  equation  (2),  the  limits 
of  the  integration  being  taken  so  as  to  include  the  entire  mass  of  the 
sphere,  and  then  find  its  differential  coefficient  with  respect  to  a. 

If  we  integrate  equation  (2)  first  with  reference  to  6,  for  which  p 
is  constant,  between  the  limits  ^  =^  0  and  d  =  2r,  we  get 


V=2.ff- 


r*  cos  <p  dr  dfp 


-% 


This  must  be  integrated  between  the  limits  ^  =  +  Jrr  and  <p  =  —  ^jtj 


22 


TIIKOHKTKAI,    ASTIIONOMY. 


l>ut  Hiuoo  ft  is  n  fuiu'tioii  of  ^f,  if  we  (litVcroutiato  the  exprosHJon  for 
(»•  with  respect  to  (f,  we  have 


and  Iience 


r  008  ip  dip  =^  —    f//», 
u 


^^-viT'-'''-''^" 


Correspondinp:  to  tlio  limits  of  tf  we  have  p~-a  —  r,  and  p~a  -\-  r; 
and  taking  the  integral  with  res[)oet  to  (t  between  thej^e  limits,  we 
obtain 

«  * 
Integrating,  finally,  between  tiie  limits  /•   •-  0  and  r  --■  r,,  we  get 

r  —        •»    „  » 


r,  being  the  radius  of  the  sphere,  and,  if  we  denote  its  entire  mass  by 

m,  this  becomes 

m 


Therefore, 


V= 


A  —  ~  '^~~  - 
da       a*' 


liilii 


from  whioh  it  appears  that  the  action  of  a  homogeneous  spherical 
mass  on  a  point  exterior  to  it,  is  the  same  as  if  the  entire  mass  were 
con<xM»trated  at  its  centre.  If,  in  the  integration  with  respect  to  r, 
we  take  the  limits  r'  and  r",  we  obtain 

and,  denoting  by  i»„  the  mass  of  a  spherical  shell  whose  radii  arc  r" 
and  r',  this  becomes 

-  a'- 

Consequently,  the  attraction  of  a  homogeneous  spherical  shell  on  a 
point  exterior  to  it,  is  the  same  as  if  the  entire  mass  were  concentrated 
at  its  centre. 

The  supposition   that  the   point  attracted   is  situated  within  a 
spherical  shell  of  uniform  density,  does  not  change  the  form  of  the 


Fl'N  DA  MKNTA  I,    I'lM  N»  1 1'l  KS. 


goiiorni  i'<|iiation ;  l)iit,  in  tlic  iiitc^nitioit  with  n'r»'n'ucc  to  y,  the 
liinit.s  will  Ix;  [»     -  r  -f  «,  iiiid  n      r  —  a,  \vlii<'li  jjive 

nii<l  this  lu-ing  iiKU'pendt'iit  of  <i,  \\v  have 

^  =  -''/-'=o. 

(Ill 

Whence  it  toHows  that  a  jxtint  phicod  in  tlie  interior  ot'  n  spherical 
shell  is  e<|ually  attracted  in  all  directions,  and  that,  il'  not  snitjeet  to 
the  action  ol'  any  cxtn'.neons  I'orce,  it  will  Im'  in  ctpiilihriuin  in  every 
position. 

7.  Whatever  may  he  the  law  of  the  chanj^c  of  the  density  of  the 
heavenly  lioilit-s  I'roni  th(>  ^nrface  to  the  centre,  we  may  rc;;ard  them 
as  composed  of  homo<;ene«.us,  eoncentrie  layers,  the  density  varyinj; 

he  indetinite.  The  action  of  each  of  <hest'  will  he  the  same  a;  if  itrJ 
mass  were  nnited  at  the  centre  of  tin'  shell  ;  and  hence  the  total  action 
of  the  hody  will  be  the  same  as  if  the  entire  mass  were  concentrated 
at  its  centre  of  j^rav'  y.  The  planets  are  indee«l  not  exactly  spheres, 
hut  oblate  spheroid>  dilferinf;  but  little  from  spheres;  and  the  error 
of  the  assumption  of  an  exact  spherical  form,  so  far  as  nilates  to 
their  action  upon  each  other,  is  extremely  small,  and  is  in  fact  com- 
pensated by  the  magnitude  of  their  distances  from  each  other;  for, 


ily  from  one  layer  to  another,  and  the  numlH-r  of  the  layers  may 


rhatc' 


be  the  f( 


)f  the  bodv,  if  its  di 


11 


,'er  may  ue  ine  lorm  oi  me  oody,  ii  its  (umen.sions  are  .- 
in  comparison  with  its  distanc(^  from  the  body  which  it  attracts,  it  is 
evident  that  its  action  will  be  sensibly  the  same  as  if  its  entire  mass 
were  concentrated  at  its  centre  of  gravity.  If  we  suppose  a  system 
of  bodies  to  be  composed  of  spherical  masses,  each  unattended  with 
any  satellite,  and  if  we  su])pose  that  the  dimensions  of  the  bodies 
are  small  in  comparison  with  their  mutual  distances,  the  formation 
of  tlie  equations  for  the  motion  of  the  bodies  of  the  system  will  be 
reduced  to  the  consideration  of  the  motions  of  simple  points  endowed 
with  forces  of  attraction  corresponding  to  the  resj)eetive  masses.  Our 
solar  system  is,  in  reality,  a  compound  system,  the  several  systems 
of  primary  and  satellites  corresponding  nearly  to  the  case  supposed  ; 
and,  before  proceeding  with  the  formation  of  the  equations  which  are 
applicable  to  the  general  case,  we  will  c  -isider,  at  first,  those  for  a 
simple  system  of  bodies,  considered  as  pf'  nts  and  subject  to  their 
mutual  actions  and  the  action  of  the  forces  which  correspond  to  the 


2t 


TII EOUETK  AL    ASTUONOM Y. 


actual  velocities  of  the  ditfcrcnt  |)art.s  of  the  system  for  any  instant. 
It  is  evident  that  we  cannot  considcT  the  nioti(<n  of  any  sinf>l(!  hody 
as  free,  and  sul)ject  only  to  the  action  of  the  ))riinitiNe  iin|)nIsion 
which  it  has  received  and  the  accelerating  foree,^  which  act  ni)on  it; 
but,  on  the  conlrary,  the  motion  of  each  ly  dy  will  de[)en(l  on  the 
force  which  acts  upon  it  directly,  and  also  on  the  reaction  due  to  the 
other  bodies  of  the  system.  The  consideration,  however,  of  the  varia- 
tions of  the  motion  of  the  several  bodies  of  the  system  is  reduced  to 
the  simple  ease  of  e(juilibrium  by  means  of  *he  general  principle  that, 
if  we  assign  to  the  diU'erent  bodies  of  the  system  motions  which  arc 
modified  by  their  mutual  action,  we  may  regard  these  motions  as 
composed  of  those  which  the  bodies  actually  have  and  of  other 
motions  which  are  destroyed,  and  which  must  therefore  necessarily 
be  such  that,  if  they  alone  existtnl,  the  system  would  be  in  oqui- 
lihrinm.  We  are  thus  enabled  to  form  at  once  the  eouations  for  the 
motion  of  a  system  of  bodies.  Let  m,  m',  m",  &(!.  be  the  masses  of 
the  several  bodies  of  the  system,  and  ;(•,  _//,  s,  x' ,  y',  z',  &c.  their  co- 
ordinates referred  to  any  system  of  rectangular  axes.  Further,  let 
the  couiponents  of  the  total  force  acting  upon  a  unit  of  the  mass  of 
VI,  or  of  the  accelerating  force,  resolved  in  directions  ])arallel  to  the 
co-ordinate  axes,  be  denoted  by  A',  V,  and  Z,  respectively,  then  will 


mX, 


m  Y, 


mZ, 


be  the  forces  which  act  upon  the  body  in  the  same  directions.  The 
velociti<  s  of  the  body  m  at  any  instant,  in  directions  parallel  to  the 
co-ordinate  axes,  will  bo 


dx 
'dt' 


dz 

di' 


and  the  corresponding  forces  arc 


m 


dx 
dt' 


m 


dy 
dt' 


dz 

''-dT 


By  virtue  of  the  action  of  the  accelerating  force,  tlu?sc  forces  for  the 
next  instant  become 


m  —.J  +  mAdf, 


m '^J^  -]- mYdt, 


wi-./  +  viZdt, 
dt 


which  may  be  written  respectively: 


MOTIOX    OF   A   SYSTEM   OF    BODIES. 


25 


dx  J  dx  ,  dx 

m    ,.  -\-  md    .    — md    .    -\-  mXdt, 


dt 


dt 


dt 


VI  ~  -4-  md  -  ■ md   ,,  +  '«  ydt, 

dt  dt  dt 

m  --77-  +  md  -,- md  —rr  +  mZdt. 

dt  dt  dt 


The  actual  vclocitios  for  this  iuotant  arc 


dx        .  dx 
"dt"^'^  dt' 


d,, 
dt 


dy 

dt' 


+  '',/. 


and  the  corrcspoiuliiig  Ibrccs  arc 

dji 


dx    ,       ,  dx 
""'dt+'^'^-Jt' 


""dt+'"'^dt' 


dz  dz 

'dt+''  df' 


dz  dz 


Comparing  these  with  the  preceding  expressions  for  the  forces,  it 

appears  that  tiie  forces  which  are  destnjyed,  in  directions  parallel  to 

the  co-ordinate  axes,  arc 

dx 


—  md    ,-  4-  mXdt, 

dt 

—  md  -J-  -j-  m  Ydt, 

dz 

—  md—T-  A-  mZdt. 

dt 


(8) 


In  the  same  manner  we  find  for  the  forces  which  will  be  destroyed 
in  the  case  of  the  body  m' : 

dx' 

—  m'd  -J-  -j-  m'X'dt, 

—  m'd^-\-m!Y'dt, 

It 

—  m'd    ~-  -|-  m'Z'dt; 

and  similarly  for  the  o>her  bodies  of  the  system.  According  to  the 
[fcneral  principle  above  ennnciatcd,  the  system  nndcr  the  action  of 
these  forces  alone,  will  be  in  c(pjilil)rinm.  The  conditions  of  e(pu- 
lihrium  for  a  system  of  points  of  invariable  but  arbitrary  form,  and 
subject  to  the  action  of  forces  directed  in  any  manner  whatever,  are 


in  which  X,,  ¥,,  Z,,  denote  the  components,  resolved  parallel  to  the 


26 


THEORETICAL  ASTRONOJfY. 


co-ordinate  axes,  of  the  forces  acting  on  any  point,  and  x,  y,  z,  the 
co-ordinates  of  the  i)oint.  These  equations  are  ecpially  applicahle  to 
the  case  of  the  equilibrium  at  any  instant  of  a  system  of  variable 
form ;  and  substituting  in  them  the  expressions  (3)  for  the  forces  de- 
stroyed in  the  case  of  a  system  of  bodies,  we  shall  have 


(I'll 
dh 


dC 
d^^ 

'  dP 


'-  —  lmY=0, 


(4) 


/    d^x  dh  \       V    ^  T         X  N       A 

"  \  ^  df  ~  ^  df  I  ~        ^-^^  ~  "^-'^  ^  ^' 

which  are  the  general  equations  for  the  motions  of  a  system  of  bodies. 

8.  Let  .1',,  iji,  Zt,  be  the  co-ordinates  of  the  centre  of  gravity  of  the 
system,  and,  by  differentiation  of  the  equations  for  the  -^o-ordinates 
of  the  centre  of  gravity,  whit.'h  are 


Imx 

X,  ■ —  -V,       ) 


we  get 


dh; 


Zm  -— 


dt' 


I  my 


^4f 


Z, 


dh, 
df 


Em  ' 


y    dh 
dfl 


Introducing  these  values  into  the  first  three  of  equations  (4),  they 
become 


IwX 


df 


df 


2m  F 


d% 
df 


ImZ 


-m 


(0) 


from  which  it  appear.^  that  the  centre  of  gravity  of  the  system  moves 
in  space  as  if  the  masses  of  the  different  bodies  of  which  it  is  com- 
posed, were  united  in  that  })oint,  and  the  forces  directly  applied  to  it. 
If  we  suppose  that  the  only  accelerating  forces  which  act  on  the 
bodies  of  the  system,  are  those  Avhich  result  from  their  mutual  action, 
we  have  the  obvious  relation  : 


mX- 


m'X', 


mY=  —  m'Y', 


mZ- 


m'Z', 


X'^ 
Hence  wt 


Ijil'i;- 


MOTION   OP   A   SYSTEM   OF    HODIE8. 

and  similarly  for  any  two  bodies ;  and,  consoquently, 


2T 


so  that  equations  (5)  become 


ImZ=0] 


de 


0, 


d^z, 
df 


=  0. 


Integrating  these  once,  and  denoting  the  constants  of  intcg>'ation  by 
c,  c',  c",  we  find,  by  combining  the  results, 


dx''  +  df  +  dz'' 
df 


c'  +  c"  +  c' 


,"2, 


and  hence  the  absolute  motion  of  the  centre  of  gravity  of  the  system, 
when  subject  only  to  the  mutual  action  of  the  bodies  which  compose 
it,  must  be  uniform  and  rectilinear.  Whatever,  therefore,  may  be 
the  relative  motions  of  the  different  bodies  of  the  system,  the  motio.i 
of  its  centre  of  gravity  is  not  thereby  aftccted. 

9.  Let  us  now  consider  the  last  three  of  eciuations  (4),  and  suppose 
the  system  to  be  submitted  only  to  the  mutual  action  of  the  bodies 
which  compose  it,  and  to  a  force  directed  toward  the  origin  of  co- 
ordinates.    The  action  of  m'  on  m,  according  to  the  law  of  gravita- 

tion,  is  expressed  by  — ,  in  which  ft  denotes  the  distance  o£m  from  vi'. 

To  resolve  this  force  in  directions  parallel  to  the  three  rectangular 
axes,  we  must  multiply  it  by  the  cosine  of  the  angle  which  the  line 
joining  the  two  bodies  makes  with  the  co-ordinate  axes  respectively, 
which  gives 


X=- 


m'  (x^  —  x) 


r_  m'  (y'  —  y) 


Z. 


m'  (/  —  z) 


Further,  for  the  components  of  the  accelei'ating  force  of  m  on  m',  we 
have 


m  {x  —  x') 


v  —  "^  <"y  —  y') 


Z' 


m  {z 


z') 


Hence  we  derive 

m  (  Yx  —  Xy)  +  m'  (  Y'x'  —  X'tf)  ^  0, 


and  generally 


(6) 


■: 


w 


28 


THEORETICAL   ASTRONOMY. 


In  a  similar  mannoi*,  Ave  find 


Im  (Xz 
2»i  (Zy  ■ 


Zx)  =  0, 

Yz)  =  0. 


(7) 


These  relations  will  not  l)e  altered  if,  m  addition  to  their  rceiproeal 
action,  the  bodies  of  the  system  are  acted  npon  by  forces  directed  to 
the  orisjin  of  co-ordinates.  Thus,  in  the  case  of  a  force  acting  npon 
m,  and  directed  to  the  origin  of  co-ordinates,  we  have,  for  its  action 
alone, 

Yx  ^  Xy,  Xz  =  Zx,  Zy  =  Yz, 

and  similarly  for  the  other  bodies.  Hence  these  forces  disappear 
from  the  erpiations,  and,  therefore,  when  the  several  bodi(\s  of  the 
system  are  subject  only  to  their  reciprocal  action  and  to  forces  directed 
to  the  origin  of  co-ordinates,  the  last  three  of  equations  (4)  become 


„    /    d'x         dh\       . 


the  integration  of  Avhich  gives 


Im  (xdy  —  ydx) 
Im  (zdx  —  xdz ) 
Imiydz  —  zdy) 


cdt, 
■■  c"dt. 


(8) 


c,  c',  and  c"  being  the  constants  of  integration.  Now,  xdy  —  ydx 
is  double  the  area  described  about  the  origin  of  co-ordinates  by  the 
projection  of  the  radius-vector,  or  line  joining  m  with  the  origin  of 
co-ordinates,  on  the  plane  of  xy  during  the  element  of  time  dt;  and, 
further,  zdx  —  xdz  and  ydz  —  zdy  are  respectively  double  the  areas 
described,  during  the  same  time,  by  the  projection  of  the  radius-vector 
on  the  planes  of  xz  and  yz.  The  constant  c,  therefore,  expresses  the 
sum  of  the  products  formed  by  midtiplying  the  arcal  velocity  of  each 
body,  in  the  direction  of  the  co-ordinate  plone  xy,  by  its  mass;  and 
c',  c",  express  the  same  sum  with  reference  to  the  co-ordinate  planes 
xz  and  yz  respectively.  Hence  the  sum  of  the  aroal  velocities  of  the 
several  bodies  of  the  system  about  the  origin  of  co-ordinates,  each 
multiplied  by  the  corresponding  mass,  is  constant;  and  the  sura  of 
the  areas  traced,  each  multiplied  by  the  corresponding  mass,  is  pro- 
portional to  the  time.     If  the  only  forces  which  operate,  are  those 


INVARIABLE   PLANE. 


29 


resulting  from  the  mutual  action  of  the  bodies  which  compose  the 
svstem,  this  result  is  correct  whatever  may  be  the  point  in  space 
taken  as  tlie  origin  of  co-ordinates. 

The  areas  described  by  the  projection  of  the  radius-vector  of  each 
body  on  the  co-ordinate  planes,  are  the  projections,  on  these  planes,  of 
the  areas  actually  described  in  space.  Wc  may,  therefore,  conceive  of 
a  resultant,  or  principal  plane  of  projection,  such  that  the  sum  of  the 
areas  traced  by  the  projection  of  each  radius-vector  on  this  plane, 
when  projected  on  the  three  co-ordinate  planes,  each  being  multiplied 
by  the  corresponding  mass,  will  be  respectively  equal  to  the  first 
luenibers  of  the  equations  (8).  Let  «,  [i,  and  y  be  the  angles  which 
tliis  principal  plane  makes  with  the  co-ordinate  planes  xy,  .vz,  and  i/z, 
respectively;  and  let  S  denote  the  sum  of  the  areas  traced  on  this 
plane,  in  a  unit  of  time,  by  the  projection  of  the  radius-vector  of 
each  of  the  bodies  of  the  system,  each  area  being  multiplied  by  the 
corresponding  mass.  The  sum  »S'  will  be  found  to  be  a  maximum, 
and  its  projections  on  the  co-ordinate  planes,  corresponding  to  the 
clement  of  time  dt,  are 


S  cos  a  dt, 


S  cos  /?  dt, 


S  cos  y  dt. 


Therefore,  by  means  of  equations  (8),  we  have 

c  =  ^  cos  a,  c'  =  *S'  cos  ,3,  c"  =  S  cos  y, 

and,  since  cos"a  +  cos^/9  +  cos^;'  =  1, 

S^  =:  C' +  C'' -^  C"\ 


Hence  we  derive 
cosa  = 


V c' -\- d' -\- c"-' 


cos/?: 


l/c'  +  C"  +  d"' 


COS  J'  = 


]/cH^''  +  c' 


"a 


These  angles,  being  therefore  constant  and  independent  of  the  time, 
show  that  this  principal  plane  of  projection  remains  constantly  par- 
allel to  itself  during  the  motion  of  the  system  in  space,  whatever 
may  bo  the  relative  positions  of  the  several  bodies ;  and  for  this 
reason  it  is  called  the  invariable  plane  of  the  system.  Its  position 
with  reference  to  any  known  plane  is  easily  determined  when  the 
velocities,  in  directions  parallel  to  the  co-ordinate  axes,  and  the 
masses  and  co-ordinates  of  the  several  bodies  of  the  system,  are 
known.     The  values  of  c,  c',  c"  are   given  by  equations  (8);  and 


30 


THE07  .ETICAL   ASTRONOMY. 


illiilti 


lionee  the  values  of  «,  ^9,  and  y,  which  determine  the  position  of  the 
invariable  plane. 

Since  the  positions  of  the  co-ordinate  ]>1anes  are  arbitrary,  we  may 
sui)j)ose  that  of  .ri/  to  coincide  with  the  invariable  plane,  which  gives 
cos  /?  -  0  and  cos  y  =  0,  and,  therefore,  c'  =^  0  and  c"  =  0.  Further, 
since  the  positions  of  the  axes  of  x  and  y  hi  this  plane  are  arbitrary, 
it  follows  that  for  every  plane  perpendicular  to  the  invariable  plane, 
the  sum  of  the  areas  traced  by  the  projections  of  the  radii-vectores 
of  the  several  bodies  of  the  system,  each  multiplied  by  th(!  corre- 
sponding mass,  is  zero.  It  may  also  be  observed  that  tiie  value  of  S 
is  constant  whatever  may  be  the  position  of  the  co-ordinate  planes, 
and  that  its  value  is  neccssni'ily  greater  than  that  of  either  of  the 
quantities  in  the  second  member  of  the  equatity. 


"2 


excc])t  when  two  of  them  are  each  equal  to  zero.  It  is,  therefore,  a 
maxinuun,  and  the  invariable  plane  is  also  the  plane  of  maximum 
areas. 

10.  If  we  suppose  the  origin  of  co-ordinates  itself  to  move  with 
uniform  and  rectilinear  motion  in  space,  the  relations  expressed  by 
equations  (8)  will  remain  unchanged.  Thus,  let  a-,,  ?y„  z,  be  the  co- 
ordinates of  the  movable  origin  of  co-ordinates,  referred  to  a  fixed 
point  in  space  taken  as  the  origin;  and  let  .r,,,  7/,,,  2,,,  x„',  yj,  zj,  etc. 
be  the  co-ordinates  of  the  several  bodies  referred  to  the  movable 
origin.  Then,  since  the  co-ordinate  planes  in  one  system  remain 
always  parallel  to  those  of  the  other  system  of  co-ordinates,  we  shall 
have 

a;  =r=  .T,  -f-  .To,  y  =  y,-\-y^,  Z  =  S,  +  2„, 

and  similarly  for  the  other  bodies  of  the  system.  Introducing  these 
values  of  x,  y,  and  z  into  the  first  three  of  equations  (4),  they  become 


"'I  rf<»  +   dC  J      ' 


0, 
0, 


The  condition  of  uniform  rectilinear  motion  of  the  movable  origin 
gives 


d\ 
di? 


0, 


•— '^  =  0 


di'      ' 


bodies,  c 


MOTION   OF  A  SOLID  BODY. 

and  the  preceding  equations  become 


31 


ImX  =  0, 


Im  '^y^  - 1 


iW 


■mr=o, 


(9) 


at' 


Substituting  the  same  values  in  the  last  throe  of  equations  (4),  ob- 
serving that  the  co-ordinates  .r,,  y,,  z,  are  the  same  for  all  the  bodies 
of  the  system,  and  reducing  the  resulting  equations  by  means  of 
equations  (9),  we  get 


2,)n  I  .r    -      - 


a-0  -  ^^.;- 1  —  -"I  (AX  —  Zi-o)       0, 
t'  I 


(10) 


»  (W 


I»i(Zy,-Yz,)=0. 


Hence  it  appears  that  the  form  of  the  equations  for  the  motion  of  the 
system  of  bodies,  remains  unchanged  when  we  suppose  the  origin  of 
co-ordinates  to  move  in  space  with  a  uniform  and  rectilinear  motion. 

11.  The  equations  already  derived  for  the  motions  of  a  system  of 
bodies,  considered  as  reduced  to  material  points,  enable  us  to  form  at 
once  those  for  the  motion  of  a  solid  body.  The  mutual  distances  of 
the  parts  of  the  system  are,  in  this  case,  invariable,  and  the  masses 
of  the  several  bodies  become  the  elements  of  the  mass  of  the  solid 
body.  If  we  denote  an  element  of  the  mass  by  dm,  the  equations  (5) 
for  the  motion  of  the  centre  of  gravity  of  the  body  become 


d% 


-jXdm, 


T  =frdm, 


d% 


=fZdm,     (11) 


the  summation,  or  integration  with  reference  to  dm,  being  taken  so  as 
to  include  the  entire  mass  of  the  body,  from  which  it  appears  that 
the  centre  of  gravity  of  the  body  moves  in  space  as  if  the  entire  mass 
were  concentrated  m\  that  point,  and  the  forces  applied  to  it  directly. 
If  we  take  the  origin  of  co-ordinates  at  the  centre  of  gravity  of 
the  body,  and  suppose  it  to  have  a  rectilinear,  uniform  motion  in 
space,  and  denote  the  co-ordinates  of  the  element  dm,  in  rofei'cnce  to 
this  origin,  by  Xq,  y^,  z,,,  we  have,  by  means  of  the  equations  (10), 


32 


TIIEORETICAI.   ASTRONOMY. 


Ill 


/( -'o  'Jl;  -  2/0    -1^^  )  dm  -/(  Kr„  -  ^„)  dm  =  0, 
/ (  ^0  ti?  -  ^0  '2^  )  dm  -f(A\  -  Zj-J  dm  =  0, 


(12) 


the  into^ratioii  Avitli  I'osju'ct  to  dm  boiiig  takon  so  as  to  include  the 
entire  iua.«s  of  the  body.  These  equations,  therefore,  determine  tiie 
motion  of  rotation  of  the  body  around  its  centre  of  tjravitv  rey-arded 
as  fixed,  or  as  having  a  uniform  rectilinear  motion  in  spacr.  Equa- 
tions (11)  determine  the  position  of  the  centre  of  gravity  for  any 
instant,  and  hence  for  the  successive  instants  at  intervals  equal  to  <lt; 
and  we  may  consider  the  motion  of  the  body  during  the  element  of 
time  (H  as  rectilinear  and  uniform,  whatever  may  be  the  form  of  its 
trajectory.  Hence,  equations  (11)  and  (12)  completely  determine  tiie 
position  of  the  body  in  s])ace, — the  former  relating  to  the  motion  of 
translation  of  the  centre  of  gravity,  and  the  latter  to  the  motion  of 
rotation  al)out  tiiis  point.  It  follows,  therefore,  that  for  any  forces 
Avhi(!h  act  upon  a  body  we  can  always  decompose  the  actual  motion 
into  those  of  the  translation  of  the  centre  of  gravity  in  space,  and  of 
the  motion  of  rotation  around  this  point;  and  these  two  motions  may 
be  considered  independently  of  each  other,  the  motion  of  the  centre 
of  gravity  being  independent  of  the  form  and  position  of  the  body 
about  this  point. 

If  the  only  forces  which  act  upon  the  body  are  the  reciprocal  action 
of  the  elements  of  its  mass  and  forces  directed  to  the  origin  of  co- 
ordinates, the  second  terms  of  equations  (12)  become  each  equal  to 
zero,  and  the  results  indicated  by  Cipiations  (8)  a])ply  in  this  case 
also.  The  parts  of  the  system  being  invariably  connected,  the  plane 
of  maximum  areas,  or  htvarkihlc  plane,  is  evidently  that  which  is 
perpendicular  to  the  axis  of  rotation  passing  through  the  centre  of 
gravity,  and  therefore,  in  the  motion  of  translation  of  the  centre  of 
gravity  in  space,  the  axis  of  rotation  remains  constantly  parallel  to 
itself.  Any  extraneous  force  which  tends  to  disturb  this  relation 
will  necessarily  develop  a  contrary  reaction,  and  hence  a  rotating  body 
resists  any  change  of  its  plane  of  rotation  not  parallel  to  itself.  We 
may  observe,  also,  that  on  account  of  the  invariability  of  the  mutual 
distances  of  the  elements  of  the  mass,  according  to  equations  (8),  the 
motion  of  rotation  must  be  uniform. 

12.  We  shall  now  consider  the  action  of  a  system  of  bodies  on  a 


MOTION   OF   A  SOLID   BODY. 


33 


distant  mass,  which  wo  will  denote  by  3f.  Let  .t,„  i/q,  Zq,  xJ,  y,/,  ;„', 
&c'.  be  the  co-ordinates  of  the  scjveral  bodies  of  the  system  rel'erred 
to  its  centre  of  gravity  as  the  origin  of  co-ordinates;  .r„  y„  and  ;, 
tlu!  co-ordinates  of  the  centre  of  {^ravity  of  the  system  referred  to 
the  centre  of  gravity  of  the  body  Jf.  The  co-ordinates  of  the  body 
5U,  of  the  system,  referred  to  this  origin,  will  therefore  be 


X  —  X,  -j-  Xq 


y=^y,  +  2/0. 


Z  =  Z,+  2o, 


and  similarly  for  the  other  bodies  of  the  system.  If  we  denote  by 
r  the  distance  of  the  centre  of  gravity  of  m  from  that  of  M,  the 
accelerating  force  of  the  former  on  an  clement  of  mass  at  the  centre 
of  gravity  of  the  latter,  resolved  parallel  to  the  axis  of  x,  will  be 


mx 


and,  therefore,  that  of  the  entire  system  on  the  element  of  3f,  resolved 

in  the  same  direction,  will  be 

„vix 


,'t  • 


We  have  also 


»•»  =  {x,  +  x,y  +  (y,  +  y,r  +  {z,  -f  z,)  \ 

and,  if  we  denote  by  r,  the  distance  of  the  centre  of  gravity  of  the 
system  from  M, 


Therefore 


r,'  ^  xi"  +  y,^  +  z^. 


3 

, J  =  (*/  +  ^0)  (^V"  +  2  {x,  x^  +  y,  2/0  +  2. 2o)  +  'V) 


We  shall  now  suppose  the  mutual  distances  of  the  bodies  of  the 
system  to  be  so  small  in  comparison  with  the  distance  r,  of  its  centre 
of  gravity  from  that  of  M,  that  terms  of  the  order  r^  may  be  neglected ; 
a  condition  which  is  actually  satisfied  in  the  case  of  the  secondary 
systems  belonging  to  the  solar  system.  Hence,  developing  the  second 
factor  of  the  second  member  of  the  last  equation,  and  neglecting  terms 
of  the  order  r^,  we  shall  have 


a  _  _«,       ^0  _  3.r,  (.r,a?„  +  y,y^  +  z,  2,,) 

„  3  "■"  „  3 


and 


r*      /•/      r?  ?•* 


Imx,, 


„mx -m 


3.r, 


'•/ 


■  {x,^mx^  +  y,^my^  -f  z,l'mz^. 


34 


TIIEOIIKTICAL    A8TUONOMY. 


But,  siiK'o  r,,,  ?/y,  -„  ar<>  the  ('(t-«>nliii:iti's  in  rrleroncc  to  the  centre  of 
gravity  of  the  system  as  origin,  we  imve 


Imxg  —  0,  i'/H^o  =  0, 

and  the  prceeding  equation  reduces  to 


-»i2o  =  0' 


'I 


In  a  similar  manner,  we  find 


y» 


Im 


,mz 


-»i 


The  second  members  of  tlicse  equations  are  the  expressions  for  the 
total  accelerating  force  due  to  the  action  of  the  bodits  if  the  system 
on  JA  resolved  })arullcl  to  the  co-ordinate  axes  respectively,  when  wo 
consider  the  several  masses  to  be  collected  at  the  centre  of  gravity 
of  the  system.  Hence  we  conclude  that  when  an  clement  of  mass 
is  attracted  by  a  system  of  bodies  so  remote  from  it  that  terms  of  the 
order  of  the  squares  of  the  co-ordinates  of  the  several  bodies,  referred 
to  the  centre  of  gravity  of  the  system  as  the  origin  of  co-ordinates, 
may  be  neglected  in  comparison  with  the  distance  of  the  system  from 
the  point  attracted,  the  action  of  the  system  will  be  the  same  as  if 
the  masses  were  all  united  at  its  centre  of  gravity. 

If  we  suppose  the  masses  m,  m',  m",  &c.  to  be  the  elements  of  the 
mass  of  a  single  body,  tlu;  form  of  the  equations  remains  unchanged; 
ar.d  hence  it  follows  that  the  mass  M  is  acted  upon  by  another  mass, 
or  by  a  system  of  bodies,  as  if  the  entire  mass  of  the  body,  or  of  the 
system,  were  collected  at  its  centre  of  gravity.  It  is  evident,  also, 
that  reciprocally  in  the  case  of  two  systems  of  bodies,  in  which  the 
mutual  distances  of  the  bodies  are  small  in  comparison  with  the 
distance  between  the  centres  of  gravity  of  the  two  systems,  their 
mutual  action  is  the  same  as  if  all  the  several  masses  in  each  system 
were  collected  at  the  common  centre  of  gravity  of  that  system;  and 
the  two  centres  of  gravity  will  move  as  if  the  m;  sses  were  thus 
united. 

13.  The  results  already  obtained  are  sufficient  to  enable  us  to  form 
the  cijuations  for  the  motions  of  the  several  bodies  which  compose  the 
solar  system.  If  these  bodies  were  exact  spheres,  which  could  be 
considered  as  composed  of  homogeneous  concentric  spherical  shells, 
the  density  varying  only  from  one  layer  to  another,  the  action  of 


MOTION  OF   A   SYSTF,>f   OF    nODIES. 


a5 


each  on  mi  oloincut  of  tlio  mass  <»f'  another  would  bo  the  same  ns  if 
llu'  entire  mass  of  the  attraetinuf  body  were  concentrated  at  its  centre 
of  f^ravity.  The  s)ij>ht  (h'viation  from  this  hiw,  arisinj^  from  the 
ellipsoidal  form  of  the  heavenly  bodies,  is  compensated  by  tlu;  majj;- 
nitiide  of  their  mntual  distances;  and,  besides,  these  mntual  distances 
are  so  great  that  the  a<'tion  of  the  attracting'  body  on  the  entire  mass 
of  the  ijody  attracted,  is  the  same  as  if  the  latter  were  concentrated 
at  its  centre  of  gravity.  Hence  the  consideration  of  the  reciprocal 
action  of  the  single  bodies  of  the  system,  is  reduced  to  that  of  material 
points  corresj)onding  to  their  respective  centres  of  gravity,  the  masses 
of  which,  however,  are  eipiivalent  to  those  of  the  corres|)()nding 
bodies.  The  mutual  distances  of  the  bodies  comjiosing  the  secondary 
systems  of  planets  attended  with  satellites  are  so  small,  in  comparison 
with  the  distances  of  the  ditt'erent  systems  from  each  other  and  from 
the  other  planets,  that  they  act  upon  these,  and  are  reci[)roeally  acted 
upon,  in  nearly  the  sanie  manner  as  if  the  masses  of  the  secondary 
systems  were  united  at  their  I'ommon  centres  of  gravity,  respectively. 
The  motion  of  the  centre  of  gravity  of  a  system  consisting  of  n 
planet  and  its  satellites  is  not  aft'ectcd  by  the  reeiiirocal  action  of  the 
bodies  of  that  system,  and  hence  it  may  be  considered  inde[)endently 
of  this  action.  The  ditference  of  the  action  of  the  other  planets  on 
a  i)lanet  and  its  satellites  will  simply  produce  inecpialitics  in  the 
relative  motions  of  the  latter  bodies  as  determined  by  their  mutual 
action  alone,  and  Avill  not  affect  the  motion  of  their  common  centre 
of  gravity.  Hence,  in  the  formation  of  the  ecj^uations  for  the  motion 
of  translation  of  the  centres  of  gravity  of  the  several  planets  or 
secondary  systems  which  compose  the  solar  system,  wc  have  sim])ly 
to  consider  them  as  points  endowed  with  attractive  forces  correspond- 
ing to  the  several  single  or  aggregated  masses.  The  investigation 
of  the  motion  of  the  satellites  of  each  of  the  planets  thus  attended, 
forms  a  problem  entirely  <listinct  from  that  of  the  motion  of  the 
common  centre  of  gravity  of  such  a  system.  The  consideration  of 
the  motion  of  rotation  of  the  several  bodies  of  the  solar  system  about 
their  respective  centres  of  gravity,  is  also  independent  of  the  motion 
of  translation.  If  the  resultant  of  all  the  forces  which  act  upon  a 
planet  passed  through  the  centre  of  gravity,  the  motion  of  rotation 
would  be  undisturbed;  and,  since  this  resultant  in  all  cases  veiy 
nearly  satisfies  this  condition,  the  disturbance  of  the  motion  of  rota- 
tion is  very  slight.  The  inequalities  thus  produced  in  the  motion 
of  rotation  are,  in  fact,  sensible,  and  capable  of  being  indicated  by 
observation,  only  in  the  case  of  the  earth  and  moon.     It  has,  indeed, 


36 


THKOliETIf'AI.   ASTUOXOMY. 


been  ri^^idly  (IciiKMistnitcd  tliiit  the  axis  of  rotation  of  tlio  oarth  rela- 
tive to  tli((  body  itsclt"  is  fixed,  so  that  the  poh'S  of  rotation  and  tho 
terrestrial  equator  preserve  constantly  the  same  posiiitm  in  reference 
to  the  surface;  and  that  also  the  velocity  of  rotation  is  c(»nstant. 
This  assures  us  of  the  permanency  of  geoffraphical  positions,  and, 
in  connection  with  the  tiict  that  the  change  of  the  lenjith  of  the 
rnean  solar  <lay  arisinj;  from  the  variation  of  the  oblicpiity  of  the 
ecliptic  and  in  the  Icuffth  of  the  tropical  year,  due  to  the  action  of 
the  sun,  nio(»n,  and  planets  upon  the  earth,  is  absolutely  insensible, 
— amountinj;  to  only  a  small  fraction  of  a  second  in  a  million  of 
years, — assures  us  also  of  the  j)erinanence  of  the  interval  whieh  we 
adopt  as  the  unit  of  time  in  astronomical  investigations. 

14.  Placed,  as  we  are,  on  one  of  the  bodies  of  the  system,  it  is 
only  possible  to  deduce  from  observation  the  relative  motions  of  the 
different  heavenly  bodies.  These  relative  motions  in  the  ease  of  the 
comets  and  primary  planets  are  referred  to  the  centre  of  the  siui, 
since  the  centre  of  gravity  of  this  body  is  near  the  centre  of  gravity 
of  the  system,  and  its  preponderant  mass  facilitates  the  integration 
of  the  e([uations.tluis  obtained.  In  the  ease,  however,  of  the  secondary 
systems,  the  motions  of  the  satellites  are  considered  in  reference  to 
the  centre  of  gravity  of  their  primaries.  We  shall,  therefore,  form 
th(^  ef|uations  for  the  motion  of  the  planets  relative  to  the  centi-e  of 
gravity  of  the  sun;  for  whit^b  it  becomes  necessary  to  consider  mt)re 
particularly  the  relation  between  the  heterogeneous  quantities,  space, 
time,  and  mass,  which  aro  involved  in  them.  Each  denomination, 
being  divided  by  the  unit  of  its  kind,  is  expressed  by  an  abstract 
number;  and  hence  it  offers  no  difliculty  by  its  j)resence  in  an  e(|ua- 
tion.  For  the  unit  of  space  we  may  arbitrarily  take  the  mean  dis- 
tance of  the  earth  from  the  sun,  and  the  mt;  n  solar  day  maybe 
taken  as  the  unit  of  time.  But,  in  order  that  Avhen  the  space  is 
expressed  by  1,  and  the  time  by  1,  the  lorc.i  or  velocity  may  also  be 
ex})resse(l  by  1,  if  the  unit  of  space  is  first  adopted,  the  relation  of 
the  time  and  the  mass — which  determines  the  measure  of  the  for(!e — 
will  be  such  that  the  units  of  both  cannot  be  arbitrarily  chosen. 
Thus,  if  we  denote  by  /  the  acceleration  due  to  the  action  of  the 
mass  m  on  a  material  point  at  the  distance  a,  and  by/'  the  accelera- 
tion corresponding  to  another  mass  7>i'  acting  at  the  same  distance, 
we  have  the  relation 


m 


TO' 


>  ' 


and  h 
be  talv 


Tntegn 
a  state 

The  ac 
measnn 
generat 
of  time 


and  hen 

W(»uld  1 

a  unit  o 

this  tiim 

in  the  ni 

wliich  tl 

will  dep 

general, 

constant 

move,  w 

if  the  fo 

tance  in 

Let  th 

to  the  foi 

and/ tilt 


and  k'  h 
mass  is,  t 
as  k-  is  c 
day  as  tli 
^vhich  wc 
<lay,  on  a 
to  the  mei 
of  tiiat  ti 


'W 


MOTION    ItKI.ATIVK   TO    IIIK   SIN. 


r.7 


and  hciK'O,  since  the  nccclcriitinn  is  propdrtioiiiil  tn  the  mass,  it  may 
1»«'  taken  as  the  measure  ot'  tiie  latter.     Hut  we  have,  for  the  measiiro 

of./; 

•^       >lt' 

Tnteffratijjji;  this,  rejijarding/as  constant,  and  the  point  to  move  from 
a  state  of  rest,  we  get 

The  acceleration  in  tiie  case  of  a  variahh'  force  is,  at  any  instant, 
measured  l»y  the  vehjcity  which  the  force  acting  at  that  instant  would 
generate,  il"  suppitsed  to  remain  constant  in  its  action,  during  a  vniit 
of  time.     The  last  e(juation  gives,  when  /  - -  i, 

and  hence  the  acceleration  is  also  measured  by  double  the  space  whirli 
would  be  described  by  a  material  point,  from  a  state  of  rest,  dui-ing 
a  unit  of  time,  the  force  being  supposed  constant  in  its  action  during 
this  tiuR',  In  each  case  the  duration  of  the  unit  of  time  is  involved 
in  the  measure  of  the  acceleration,  and  hence  in  that  of  the  mass  on 
which  the  acceleration  depends;  and  the  unit  of  mass,  or  of  the  Ibree, 
will  depend  on  the  duration  which  is  chosen  for  the  unit  of  time.  In 
general,  therefore,  wc  regard  as  the  unit  of  mass  that  which,  acting 
constantly  at  a  distance  c(pial  to  unity  on  a  material  ])oint  free  to 
move,  will  give  to  this  point,  in  a  unit  of  time,  a  velocity  which, 
if  the  force  ceased  to  act,  would  cause  it  to  describe  the  unit  of  dis- 
tance in  the  unit  of  time. 

Let  the  unit  of  time  be  a  mean  solav  day;  A^  the  ac(!eleration  due 
to  the  force  exerted  by  the  mass  of  the  sun  at  the  unit  of  distance; 
and /the  acceleration  corresponding  to  the  distance  /•;  then  will 

and  Jc^  becomes  the  measure  of  the  mass  of  the  sun.  The  unit  of 
mass  is,  therefore,  equal  to  the  mass  of  the  sun  taken  as  many  times 
as  Ic-  is  contained  in  unity.  Hence,  when  we  take  the  mean  solar 
day  as  the  unit  of  time,  the  mass  of  the  sun  is  measured  by  k';  by 
Avhieh  we  arc  to  understand  that  if  the  sun  acted  during  a  mean  solar 
day,  on  a  material  point  free  to  move,  at  a  distance  constantly  equal 
to  the  mean  distance  of  the  earth  from  the  sun,  it  would,  at  the  end 
of  that  time,  have  coramunieated  to  the  point  a  velocity  which,  if 


38 


Til KOUKTK  A  I,    ASTIU ).\()M Y. 


Ili:^ 


the  Wn'va  did  not  tlicrcai'tcT  iu'f,  would  vixusv.  it  to  doscriho,  in  a  unit 
o'"  time,  tiu'  spaco  i:X|)n'ss((l  l»y  Ir. 

Tlic  a<'('(!U'ratiou  duf  to  tiio  action  of  the  sun  at  the  unit  of'distanc*.' 
is  designated  by  Ir,  since  the  scjiian;  root  of  this  (juantity  ai>[)ears 
frer|uently  in  t\w  lornuda)  wliich  will  be  derived. 

H'  \V(i  take  arbitrarily  the  mass  of  the  sun  as  th((  unit  of  mass  the 
unit  of  time  must  be  (hiferniined.  Let  /  denote  the  luimber  of  mean 
solar  days  which  nuist  i)e  taken  for  the  nuit  of  time  when  tlu;  unit 
oi'  mass  is  tiu  mass  of  the  sun.  Tiie  space  which  the  fon^e  (hie  to 
this  n.ass,  actinjj;  constantly  on  a  material  point  ut  a  distance  e(jual  to 
the  mean  distance*  of  the  earth  from  the  sun,  would  clause  the  point 
to  describe  in  the  time  /,  is,  accordin<>;  to  ecpiation  (l-'J), 

But,  since  t  ex[)re!ises  the  nund)er  of  mean  solar  days  in  the  unit  of 
time,  the  measure  of  the  acceleration  corresponding  tu  this  unit  is  2.s, 
and  this  being  the  unit  of  force,  we  have 


and  henee 


kH^  ==  1 ; 
1 


t  — . 


ii'l 


i  ill  1'!'^ 


Therefore,  if  the  mass  of  the  sun  is  r<>garded  as  the  unit  of  mass,  the 
number  of  n»eau  solar  days  in  the  unit  of  time  will  be  equal  to  unity 
divided  by  the  s((nare  root  of  the  acceleration  due  to  the  force  exerted 
by  this  mass  at  the  unit  of  flistance.  The  munericd  value  of  h  will 
be  subsefpiently  found  in  be  <).()172()21,  wliich  gives  58.1.'}244  mean 
sohir  days  for  the  unit  of  time  when  the  inas;S  of  the  sun  is  taken  as 
the  unit  of  mass. 

15.  JiCt  .)•,  I/,  z  he  the  co-ordinates  of  a  lujavenly  body  referred  to 
tlie  centre  of  gravity  of  the  sun  as  the  origin  of  co-ordinates;  /•  its 
radiiiK-vccfor,  or  distance  from  this  origin;  and  let  m  denote  the 
quotien*^  obtained  by  dividing  it'i  ma.>-s  by  that  of  the  sun;  tluMi, 
taking  the  mean  solar  day  as  the  unit  of  lime,  the  mass  of  \\u'  sun  is 
oxj»rcsse<l  by  k^,  and  that  of  the  i)lanet  or  (lomet  by  vik'.  For  a 
second  body  let  the  co-ordinates  be  .<•',  //',  z' ;  the  distance  from  the 
sun,  /•';  and  the  mass,  vi'lc';  and  similarly  for  the  other  bodies  of  the 
system.  Let  the  co-ordinates  of  the  centre  of  gravity  of  the  sun 
referred  to  any  nxed  j)oint  in  space  be  ?,  jj,  ^,  the  co-ordinate  planes 
being  parallel  to  those  of  x,  y,  and  s,  respectively;    then  will  the 


arOTION    UELATIVi:   TO   TIIK   SUX. 


accolcration  due  to  the  ju-tion  of  vi  on  the  .sun  he  expressed  hy     -■» 

an<l  the  three  components  of  this  foree  in  direetions  parallel  to  tho 
co-ordinate  axes,  n^spectively,  will  be 


-a  * 

VIKT  — , 


i«P-^ 


»iP  — 


The  action  of  m'  on  the  sun  will  he  expressed  by 


m'k' 


,«'P  ■" 


/J' 


va' 


..'3 


and  henc(>  the  acceleration  due  to  the  condn'ned  and  sinudtanc^us 
action  of  the  several  bodies  of  the  system  on  the  sun,  resolveil  par- 
allel to  the  co-ordinate  axes,  will  be 


/• 


■  >.'".'/ 


^••^2' 


,W2 


The  motion  of  the  centre  of  gravity  of  the  sun,  relative  to  tho  fixed 
origin,  will,  therefore,  be  detern)ined  by  the  equations 


^,  mx 


A         ,.,v'".'/ 


/t'2''"'-.    (14) 


Let  f)  denote  the  distance  of  m  from  to';  //  its  distance  fron.  /;(", 
adding  an  accent  for  eacii  suc:;essive  body  considered ;  then  will  the 
action  of  the  boilies  /u',  la",  tte.  on  m  be 


•red  to 

/•  its 

)te  the 

then, 

sun  is 

l^'or  a 

oni  the 

of  the 

W    SlUl 

jdanes 
■ill  the 


K  -        -, 

of  which  th(!  three  components  p;iraiiel  to  the  co-ordinate  axes,  re- 
spectively, are 


yr,2V../ 


)n 


ic^^n^y-^, 


k'l',"'^ 


■III 


The  action  of  the  sun  on  vi,  resolve;!  in  the  same  man'U'r,  is  expressed 

by 

k\v  khj  p2 


v 


r" 


which  arc  negative,  since  the  force  tends  to  diminish  the  co-ordinates 
a,  y,  and  s.  The  three  cf)mponents  of  the  total  action  of  the  other 
bodies  of  the  system  ou  vi  are,  therelbrc, 


40 


THEOKETICAL   ASTRONOMY. 


vl!l 


.11 1 ! 


m'  (x'  —  x) 

—^3 


+  ^'-"  .,3 


-,       f-  K  -  i  ■, 


and,  since  the  co-ordinates  of  m  referred  to  the  fixed  origin  are 

f  +  a;,  7i-\-y,  C  +  2, 

the  equations  which  determine  the  absohite  motion  are 


-"^  ^  rf<»  ^ 


^ 

r* 


ifc^i: 


m'  (a;'  —  a;) 


rf<' 


(15) 


the  symbol  of  summation  in  the  second  members  relating  simply  to 
the  masses  and  co-ordinates  of  the  several  bodies  which  act  on  m, 

exclusive  of  the  sun.     Substituting  for  -^,  -r^,  and  ~-  their  values 

given  by  equations  (14),  we  get 


de  ^     ^  1*  \ 


y 


(16) 


Since  x,  y,  z  are  the  co-ordinates  of  m  relative  to  the  centre  of  gravity 
of  the  sun,  these  equations  determine  the  motion  of  m  relative  to  that 
point.  The  second  members  may  be  put  in  another  form,  which 
greatly  facilitates  the  solution  of  some  of  the  problems  relating  to 
the  motion  of  m.     Thus,  let  us  put 


n  _  J«L  /  1      xx'^-yy'-]-zz'  \        m"    /  1 
i  +  m\p  r"  f'^l-{-m\p' 


to"    /  1      xx"+  yy"+  zz" 


»"3 


J+....&C., 

(17) 


and  we  shall  have  for  the  partial  differential  coefficient  of  this  with 
respect  to  x, 

\   dp^ 

■"'  dx       r 


\dxj      l+m\       p''dx       r"'/'^l-j-TO\      p"  dx       ,-'"  /  "^  " ' '   "' 


MOTION   RELATIVE  TO  THE  SUN. 


41 


But,  since 


v;c  have 


P'  =  (x'  -  xY  +  (2/  -  y)'  +  (/  -  zy, 
P"=  ix"  -  xf  +  {f  -^  yy  +  (z"  -  zy, 


dx 


dp!_ 
dx 


a!'—x 


and  hence  we  derive 

/(?fl\_     ?ft'     laS  —  x        x'  \         m"    lx"—x       x"  \ 


+ 


*&c. 


We  1^  a  also,  in  the  same  manner,  for  the  partial  differential  coeffi- 

r\rm.-,  \,' ;'i  respcct  to  y  and  z, 

The  cc^uations  (16),  therefore,  become 

*:  +  ..(! +„.), 4  =  .. (I +»)('!). 


(18) 


;J,H-^ni+m)i.  =  P(l    1  «0(f)< 


It  will  ^ 


u  uo  '<si '  \  i-l  I. '.fit  the  second  members  of  equations  (16)  ex- 
press the  differenct.  b''^  een  the  action  of  the  bodies  m',  m",  &c.  on 
m  and  on  the  smi,  resoived  para'lcl  to  <^bo  co-ordinate  axes  resji  act- 
ively. The  mutual  distances  of  the  })1.  els  are  such  that  these  qi  an- 
titics  are  generally  very  small,  and  wo  .lay,  therifore,  in  a  irst 
approximation  to  the  motion  of  m  relative  to  the  sun,  neglect  the 
second  members  of  these  equations;  and  the  intc<rrals  which  may 
then  be  derived,  express  what  is  ci  lied  the  undisturbed  motion  of  vi. 
By  means  ^i.  lio  results  thus  obtained  for  the  several  bodies  succes- 
sively, the  fi,'  •  \  pvitn  values  of  the  second  members  of  equations 
(16)  ma/  be  ioi.nd,  and  hence  a  still  closer  apj^roximation  to  the 
actual  motion  of  m.  The  force  whose  components  are  expressed  by 
the  second  n. embers  of  these  equations  is  called  the  disturbinr/ force; 


;!.i 


42 


THEORETICAL   ASTRONOMY. 


ami,  usin<2;  tlio  second  form  of  the  c([iiations,  the  fimction  Q,  Avhich 
determines  these  components,  is  eaUed  thj  2'"''"^«''^'"i//'""'''o».  The 
complete  solution  of  the  problem  is  facilitated  by  an  artifice  of  the 
infinitesimal  calculus,  known  as  the  variation  of  parameters,  or  of 
constants,  according  to  which  the  complete  integrals  of  equations  (16) 
are  of  the  same  form  as  those  obtained  by  putting  the  second  mem- 
bers equal  to  zero,  the  arbitrary  constants,  however,  of  the  latter 
integration  being  regarded  as  variables.  These  constants  of  integra- 
tion arc  the  dcmcntH  which  determine  the  motion  of  m  relative  to  the 
sun,  and  when  the  disturbing  force  is  neglected  the  elements  are  pure 
constants.  The  variations  of  these,  or  of  the  co-ordinates,  arising 
from  tiie  action  of  the  '\  ^^'irl^ing  force  are,  in  almost  all  cases,  very 
small,  and  are   called  t.l.  'iirbations.     The  problem  which  first 

presents  itself  is,  therefore,  i  determination  of  all  the  (iircumstances 
of  the  undisturbed  motion  of  the  heavenly  bodies,  after  which  the 
action  of  the  disturbing  forces  may  be  considered. 

It  may  be  further  remarked  that,  in  the  formation  of  the  preceding 
equations,  we  have  supposed  the  different  bodies  to  be  free  to  move, 
and,  therefore,  subject  only  to  their  mutual  action.  There  are,  in- 
deed, facts  derived  from  the  study  of  the  motion  of  the  comets  which 
seem  to  indicate  that  there  exists  in  space  a  residing  medium  which 
opposes  the  free  motion  of  all  the  bodies  of  the  system.  If  such  a 
medium  actually  exists,  its  effect  is  very  small,  so  that  it  can  be  sen- 
sible only  In  the  case  of  rare  and  attenuated  bodies  like  the  comets, 
since  the  accumulated  observation',  of  the  different  planets  do  not 
exhibit  any  effect  of  such  resistance.  But,  if  we  assume  its  existence, 
it  is  evidently  necessary  only  to  add  to  the  second  members  of  equa- 
tions (IG)  a  force  which  shall  represent  the  effect  of  this  resistance, — 
which,  therefore,  becomes  a  part  of  the  disturbing  force, — and  the 
motion  of  m  will  be  completely  determined. 

IG.  When  we  consider  the  undisturbed  motion  of  a  planet  or 
comet  relative  to  the  sun,  or  simply  the  motion  of  the  body  relative 
to  the  sun  as  subject  only  to  the  reciprocal  action  of  the  two  bodies, 
the  equations  (IG)  become 

g+j'a  +  .»)^=o, 


(19) 


•11 


MOTION   RELATIVE   TO   THE  SUN. 


43 


The  equations  for  the  undisturbed  motion  of  a  satellite  rclati\  o  to  its 
primary  are  of  the  same  form,  the  value  of  k',  however,  being  in  this 
casie  the  acceleration  due  to  the  force  exerted  by  the  mass  of  the 
primary  at  the  unit  of  distance,  and  in  the  ratio  of  tiie  mass  of  the 
satellite  to  that  of  the  primary. 

The  integrals  of  these  equations  introduce  six  arbitrary  constants 
of  integration,  which,  when  known,  will  completely  determine  the 
undisturbed  motion  of  m  relative  to  the  sun. 

If  we  multiply  the  first  of  these  equations  by  y,  and  the  second  by 
X,  and  subtract  the  last  product  from  the  fi'-^t,  we  shall  find,  by  inte- 
grating the  result, 

xdy  —  ydx 


dt 


c  being  on  arbit-ary  constant. 
In  a  similar  manner,  we  obtain 


=  c. 


xdz  —  zdx 
dt 


c', 


dt      " ""  • 


If  we  multiply  these  three  equations  respectively  by  z,  —  y,  and  x, 
and  add  the  products,  we  obtain 

cz  —  c'y  -\-  c"x  =  0. 

This,  being  the  equation  of  a  plane  passing  through  the  origii.  of 
co-ordinates,  shows  that  the  path  of  the  body  relative  to  the  sun  is  a 
plane  curve,  and  that  the  plane  of  the  orbit  jmsscs  through  the  centre 
of  the  sun. 

Again,  if  we  multiply  the  first  of  equations  (19)  by  2d.v,  the  second 
bv  2dy,  and  the  thii'd  by  2dz,  take  the  sum  and  integrate,  we  shall 
find 

dl+JpJt  +  2^'(1  +  ,n)f-!:^!:!^-±^ : 


0. 


But,  since  r' =  x^  -{-  y^  -\-  sr,  we  shall  have,  by  diiferentiation, 

rdr  =  xdx  +  ydy  +  ^^z. 
Therefore,  introducing  this  value  into  the  preceding  equation,  we  obtain 

a.r^'+df  +  dz'      2F(l  +  m) 


dt' 


+  A  =  0, 


(20) 


i;i 


'«! 


n 


h  being  an  arbitrary  constant. 


44 


THEORETICAL  ASTRONOMY. 


;!■:  ■ji 


If  wc  add  together  the  squares  of  llin  expressions  for  c,  c',  and  c", 
and  put  (?  +  a'^  +  c"'^  =  4/^,  we  shall  have 


f.r'  +  f  +  2^)  (fh^  +  rfy»  4-  dz^)       {xdx  +  ?/<?/  +  zdzf 


or 


cif^ 

df 

'             dt' 

r'dr' 

de  ~ 

-4/^ 

4/; 


(21) 


If  we  represent  by  dv  the  infinitely  small  angle  contained  between 
two  consecutive  radii-vectores  r  and  r  +  dr,  since  dx-  +  dy'^  +  c/r  is 
the  square  of  the  element  of  path  described  by  the  body,  wc  shall 

have 

dx"  +  dy""  +  dz""  =  dr''  +  r'dv''. 


Substituting  this  value  in  the  preceding  equation,  it  becomes 

rVy  =  2fdt. 


(22) 


The  quantity  rhlv  is  double  the  area  included  by  the  element  of  path 
described  in  the  element  of  time  df,  and  by  the  mdii-vectores  r  and 
r  +  dr;  and/,  therefore,  represents  the  areal  velocity,  which,  being  a 
constant,  shows  that  the  radius-vector  of  a  planet  or  comet  describes 
equal  areas  in  equal  intervals  of  time. 

From  the  equations  (20)  and  (21)  we  find,  by  elimination. 


dt 


rdr 


(23) 


V':..-k\l-irm)  —  hr'—Af' 
Substituting  this  value  of  dt  in  equation  (22),  we  get 

ry  2rlc'  (1  +  Hi)  —  hr'—  4/» 
which  gives,  in  order  to  find  the  maximum  and  minimum  values  of  /•, 

dr  __  rl/2rF(l  -j-  m)  —  hr'  —  Af^ 


(24) 


or 

Therefore 

and 


dv  ~~  2/  ~"  ^' 

2rP  (1  +  "0  —  /'  '•'  —  4/'  =  0. 

k\l+m)         I      4f       IHlTmr 
h         "^  V         h    "^  h' 

F(l+m) 


_J_i(l  +  ^'^l  +  "^)' 


h  ^         h     '  h' 

are,  respectively,  the  maximum  and  minimum  values  of  /•.    The 


! 


j)oints  of 
rcspondiii 
the  aphel 
values,  re 

ill  which  ^ 

(24),  it  be 


the  integri 


w  oeins:  ai 


from  whic 


which  is  tl 
focus,  p  be 
angle  at  tli 
[)lane  of  t 
axis  a. 

If  the  a 
and 


The  ang 
Hence  \ 
around  the 
Observatioi 
usually  of 
ellipses  of 
oumstance  i 


MOTION    RKI.ATIVE  TO  THE  SUN. 


45 


j)()ints  of  the  orbit,  or  trajectory  of  the  body  relative  to  the  sun,  cor- 
rt'sj)0ii(ling  to  these  values  of  /,  are  called  the  (ij)fiides;  the  former, 
the  aphcfion,  and  the  latter,  the  perihelion.  If  we  represent  these 
values,  respectively,  by  a(l  +  e)  and  (((1  —  c),  we  shall  ■•uve 


h  = 


P(l  +  "0 


4/'  =  ak'  (1  -f  m)  (1  —  e')  =  k'j)  (1  +  m), 


in  whichp  =  a  (1 — e^).     Introducing  these  values  into  the  erjuation 
(24),  it  becomes 


dv  = 


\/  p  dr 


P-dl 
e      r 


-P 


(^^-^)■ 


the  integral  of  which  gives 


-\  1  ( p 


lo  -f-  cos     — 


(f-')' 


10  being  an  arbitrary  constant.     Therefore  we  shall  have 

7(7-l)  =  cos(«-«>), 


from  which  we  derive 


r  = 


P 


1  +  e  cos  (y  —  w)' 


wliich  is  the  polar  equation  of  a  conic  section,  the  pole  being  at  the 
focus,  p  being  the  semi-parameter,  e  the  eccentricity,  and  v  —  id  the 
angle  at  the  focus  between  the  radius-vector  and  a  fixed  line,  in  the 
plane  of  the  orbit,  making  the  angle  to  with  the  semi-transverse 
axis  a. 

If  the  ang  ,v  —  co  \%  counted  from  the  perihelion,  we  have  fo'  =  0, 
and 

P 


\  -\-  e  cos  V 


(25) 


The  angle  v  is  called  the  true  anomaly. 

Hence  we  conclude  that  the  orbit  of  a  heavenly  body  revolvinr/ 
around  the  sun  is  a  conic  section  with  the  sun  in  one  of  the  foci. 
01)servation  shows  that  the  planets  revolve  around  the  sun  in  ellipses, 
usually  of  small  eccentricity,  while  the  comets  revolve  either  in 
ellipses  of  great  eccentricity,  in  parabolas,  or  in  hyperbolas,  a  cir- 
cumstance which,  as  >\e  shall  have  occasion  to  notice  hereatter,  greatly 


4Q 


TiiEonirncAL  astkonojiy. 


lessons  the  amount  of  labor  in  many  computations  respecting  their 
motion. 

Introducing  into  equation  (23)  the  values  of  h  and  4/-  alrcadv 
found,  we  obtain 

, l/« rdr 


which  may  be  written 


or 


dt^ 


ky'l  +  m^ 


^^-m     yj^-i-)- 


the  integration  of  which  gives 


t 


ki/ 


f^(--(''-~)-«V'-("-'f)+«  (2<i)  I  ™''™« 


In  the  perihelion,  r  -=  a  (1  —  e),  and  the  integral  reduces  to  t'  =  C; 
therefore,  if  we  denote  the  time  from  the  perihelion  by  (q,  we  shall 
have 


''-^^^nh-T-^h^yj^-i'^)')-     (^ 


7) 


In  the  aphelion,  r  =--a{l-he);  and  therefore  we  shall  have,  for  the 
time  in  which  the  body  passes  from  the  i>erihelion  to  the  aphelion, 

3 


2^  = 


--^rrrr  TT, 


ky'l  -\-  VI 


T  being  the  periodic  time,  or  time  of  one  revolution  of  the  planet 
around  the  sun,  a  the  semi-transverse  axis  of  the  orbit,  or  mean  dis- 
tance  from  the  sun,  and  t:  the  semi-circumference  of  a  circle  whose 
radius  is  unity.     Therefore  we  shall  have 


■An" 


k'  (1  +  my 


(28) 


MOTION   REr.ATIVK  TO  THE  SUN. 


47 


For  a  second  planet,  we  shall  have 

'  -4'.     ,j 


J^  (1  +  m')  ' 

and,  consequently,  between  the  mean  distances  and  periodic  times  of 
any  two  planets,  we  have  the  relation 


(1  4^)1)7^ 

(1  +  j^'K'^ 


(29) 


If  the  masses  of  the  two  planets  in  and  m'  are  very  nearly  the 
same,  we  may  take  \-\-m=^l-\-  m' ;  and  hence,  in  this  oxxi^^,  it  follows 
that  the  HquarcH  of  the  periodic  times  arc  to  each  other  as  the  cuIhv  of 
the  mean  iJistanccfi  from  the  ,mn.  The  same  result  may  bo  stated  in 
another  form,  which  is  sometimes  mox'c  convenient.  Thus,  since  rrah 
is  the  area  of  the  ellipse,  a  and  b  representing  the  semi-axes,  we 
shall  have 

- —  =/=-  areal  velocity; 


and,  since  P  =  a^  (1  —  c^),  Ave  have 


/= 


SI                     1  4      — 

Kcr  a-  (1  —  e")  ^ Tta^  ^/p 


which  becomes,  by  substituting  the  value  of  r  already  found. 


f=:Ul/p(l+Vl). 

In  like  manner,  for  a  second  planet,  Ave  have 


(30) 


/'==U/y(l+»0; 

and,  if  the  masses  are  such  that  avc  may  take  1  -\-  m  sensibly  equal 
to  1  +  wi',  it  folloAVS  that,  in  this  case,  the  areas  described  in  equal 
times,  in  different  orbits,  are  proportional  to  the  square  roots  of  their 
parameters. 

17.  We  shall  noAV  consider  the  signification  of  some  of  the  con- 
stants of  integration  already  introduced.  Let  i  denote  the  inclination 
of  the  orbit  of  m  to  the  plane  of  xi/,  A\diich  is  thus  taken  as  the  plane 
of  reference,  and  let  ^  be  the  angle  formed  by  the  axis  of  x  and  the 
line  of  intersection  of  the  plane  of  the  orbit  Avith  the  plane  of  xy; 
then  Avill  the  angles  i  and  SI  determine  the  position  of  the  plane  of 


48 


THKORfrrirAL  astkoxomy. 


li  ::|v  <    .i:i; 


the  orbit  in  ^pace.  The  constants  c,  c',  and  c",  involved  in  the 
equation 

cz  —  c'y  -1-  c"«  =  0, 

are,  res])octivc}y,  doul)lo  the  projections,  on  tlic  co-ordinate  planes, 
xy,  xz,  and  yz,  of  the  areal  velocity/;  and  iience  we  shall  have 

c  =  2/  cos  i. 

The  projection  of  2/  on  a  plane  passing  through  the  intersection  of 
the  plane  of  the  orbit  with  the  plane  of  xy,  and  perpendicular  to  the 
latter,  i8 

2/sini; 

and  the  projection  of  this  on  the  plane  of  xz,  to  which  it  is  inclined 
at  an  angle  equal  to  Si,  gives 

c' =  2/sinicos  ft. 

Its  projection  on  the  plane  of  yz  gives 

c"  =  2/sintsin  ft. 


Hence  we  derive 

a  cos  i  —  y  sin  i  cos  ft  +  *'  sin  i  cm  ft  =  0, 


(31) 


which  is  the  equation  of  the  plane  of  tlie  orbit;  and,  by  means  of 
the  value  of  /  in  terms  of  p,  and  the  values  of  c,  c',  c",  wo  derive, 
also, 

dx 


^~dt~  ^•''Tt  ^  ^"^^  (1  +"0  cosi, 


dz 
dz 


dx 

z-jr  =  k  i/p  (1  -}-  m)  cos  ft  sin  i, 


(32) 


dy 

it 


=  k  \/p  (1  +  m)  sin  ft  sin  i. 


These  equations  will  enable  us  to  determine  ft ,  i,  and  j9,  when,  for 
any  instant,  the  mass  and  co-ordinates  of  m,  and  the  comj^onents  of 
its  velocity,  in  directions  i)arallel  to  the  co-ordinate  axes,  are  known. 
The  constants  a  and  c  are  involved  in  the  value  of  ji,  and  hence  four 
constants,  or  dements,  are  introduced  into  these  equations,  two  ot 
which,  a  and  e,  relate  to  the  form  of  the  orbit,  and  two,  ft  and  ^,  to 
the  position  of  its  plane  in  space.  If  we  measure  the  angle  v  —  m 
from  the  point  in  which  the  orbit  intersects  the  plane  of  xy,  the  con- 
stant io  will  determine  the  position  of  the  orbit  in  its  own  plane. 
Finally,  the  constant  of  integration  C,  in  equation  (26),  is  the  time 


of  passage 
of  thf  bo(i 
undisiurbi 
Let  V  i 
equation  (; 


At  the  pe 

equation,   \ 

aphelion,  1 

In  the  p 


which  will 
It  will  be  c 
of  r,  in  an 
since  n  is 
orbit  is  still 
the  velocit}- 
direction  of 
described. 

If  the  po 
and  magniti 
will  enable 
But  since  v 
of  the  prim 
the  aid  of 
therefore,   b 
unknown  el 
gate  those  a 
centric  and 
be  known, 
the  problem 
by  observati 

18.  To  de 
system,  we  h 


MOTION    UKLATIVE  TO  THE  SUN. 


49 


of  pnpsnpjo  through  the  perihelion;  and  this  dctornuncs  tho  position 
of  thf  body  in  its  orbit.  When  these  six  constants  arc  known,  tho 
undisturbed  orbit  of  the  body  is  completely  deternnned. 

Let  V  denote  the  velocity  of  tho  body  in  ita  orbit;  then  will 
equation  (liOj  become 

At  tho  perihelion,  r  is  a  minimum,  and  hence,  according  to  this 
equation,  the  corresponding  value  of  V  is  a  maximum.  At  the 
aphelion,  V  is  a  minimum. 

In  tho  parabola,  a  =  oo,  and  hence 


V=kVl  +m^-, 


which  will  determine  tho  velocity  at  any  instant,  when  r  is  known. 
It  will  be  observed  that  the  velocity,  corresponding  to  the  same  value 
of  )•,  in  an  elliptic  orbit  is  less  than  in  a  parabolic  orbit,  and  that, 
since  a  is  negative  in  the  hyperbola,  the  velocity  in  a  hyj)crboHc 
orbit  is  still  greater  than  in  the  case  of  the  parabola.  Further,  since 
the  velocity  is  thus  found  to  be  independent  of  the  eccentricity,  the 
direction  of  the  motion  has  no  influence  on  tho  species  of  conic  section 
described. 

If  the  position  of  a  heavenly  body  at  any  instant,  and  the  direction 
and  magnitude  of  its  velocity,  arc  given,  tho  relations  already  derived 
will  enable  us  to  determine  the  six  constant  elements  of  its  orbit. 
But  since  we  cannot  know  in  advance  the  magnitude  and  direction 
of  tho  primitive  impulse  comnuinicated  to  the  body,  it  is  only  by 
the  aid  of  observation  that  these  elements  can  be  derived;  and 
therefore,  l)efore  considering  the  formula)  necessary  to  deteruiine 
unknown  elements  by  means  of  observed  positions,  we  will  investi- 
gate those  which  are  necessary  for  the  determination  of  the  helio- 
centric and  geocentric  places  of  the  body,  assuming  the  elements  to 
be  known.  The  results  thus  obtained  will  facilitate  the  solution  of 
the  problem  of  finding  the  unknown  elements  from  the  data  furnished 
by  observation. 

18.  To  determine  the  value  of  k,  which  is  a  constant  for  the  solar 
system,  we  have,  from  equation  (28), 


ifc  = 


2jr 


a 


1/1  + 


m 


to 


THKOUKTICAT.   ASTRONOMY. 


Tn  the  case  of  the  earth,  «  ==  1,  and  therefore 

2rr 


tkI  4-  TO 

In  rcdueing  this  formula  to  numbers  we  shouUl  properly  use,  for  r, 
the  alisolute  length  of  the  sidereal  year,  whieh  is  invariable.  The 
ctfet't  of  the  aetion  of  the  other  bodies  of  the  system  on  tiie  earth  is 
to  produee  a  very  small  secular  change  in  its  mean  longitude  corre- 
sponding to  any  fixed  date  taken  as  the  epoch  of  the  elements;  and 
a  correction  corresponding  to  this  secular  variation  should  be  apj)lied 
to  the  value  of  t  derived  from  observation.  The  effect  of  this  cor- 
rection is  to  slightly  increase  the  observed  value  of  r;  but  to  deter- 
mine it  with  precision  recpiires  an  exact  knowledge  of  the  masses  of 
all  the  bodies  of  the  system,  and  a  complete  theory  of  their  relative 
motions, — a  j)roblem  which  is  yet  incompletely  solved.  Astronomical 
usage  has,  therefore,  sanctioned  the  employment  of  the  value  of  k 
found  by  means  of  the  length  of  the  sidereal  year  derived  directly 
from  observation.  This  is  virtually  adopting  as  the  unit  of  space  a 
distance  which  is  very  little  less  than  the  absolute,  invariable  mean 
distance  of  the  earth  from  the  sun;  but,  since  this  unit  may  be  arbi- 
trarily chosen,  the  accuracy  of  the  results  is  not  thereby  affected. 

The  value  of  t  from  which  the  adopted  value  of  k  has  been  com- 
puted, is  365.2563835  mean  solar  days;  and  the  value  of  the  com- 
bined mass  of  the  earth  and  moon  is 


for  the  exp 
Since,  in 

it  will  be  ( 

tuotion  of  1 
In  the  n 

that  it  may 

have 


For  the  old 
piession  (3? 

19.  Let  I 
being  at  the 


If  we  reprc! 
and  a  line 
shall  have 


m  = 


354710" 


'II 


Hence  we  have  log  r  =  2.5625978148 ;  log  i/l  +  m  ==  0.0000006122 ; 
log  2;:  =  0.7981798684;  and,  consequently, 

log  ;fc  =  8.2355814414. 

If  wo  multiply  this  value  of  k  by  206264.81,  the  number  of  seconds 

of  arc  corresponding  to  the  radius  of  a  circle,  we  shall  obtain  its 

value  exp'^essed  in  seconds  of  arc  in  a  circle  whose  radius  is  unity,  or 

on  the  orbit  of  ^he  earth  supposed  to  be  circular.     The  value  of  k  in 

seconds  is,  therefore, 

log /t  =  3.5500065746. 

The  quantity  —  expresses  the  mean  angular  motion  of  a  planet 

in  a  meat   solar  day,  and  is  usually  designated  by  fi.    We  shall, 
therefore,  have 


and,  since  c 
which  we  ha 


The  angle  <p 
Again,  sic 


It  is  evident, 
elliptic  orbit 
of  r  correspo 
to  the  apheli 
helion. 


mi:!! 


MOTION   RELATIVE  TO  THE  HUN. 
/fcl/r  -f-  n 


61 

(33) 


I'or  the  expression  for  the  mmn  fhtily  motion  of  a  plunet. 

Since,  in  the  case  of  the  earth,  Vl  +  m  tlitters  very  little  from  1, 
it  will  be  observed  that  k  very  nearly  expresses  the  mean  angular 
motion  of  the  earth  in  a  mean  solar  day. 

In  the  ease  of  a  sniall  planet  or  of  a  eomet,  the  mass  m  is  so  small 
that  it  may,  without  sensible  error,  be  neglected;  and  then  we  shall 
have 


For  the  old  planets  whose  masses  are  considerable,  the  rigorous  ex- 
piession  (33)  must  be  used. 

19.  Let  us  now  resume  the  polar  equation  of  the  ellipse,  the  polo 
being  at  the  focus,  which  is 


r  = 


a(l  — e') 
1  +  e  cos  V 


If  we  represent  by  <p  the  angle  included  between  the  conjugate  axis 
and  a  line  drawn  from  the  extremity  of  this  axis  to  the  focus,  we 
shall  have 

sin  ^  =  e; 

and,  since  a(l — e")  is  half  the  parameter  of  the  transverse  axis, 
which  we  have  designated  by  p,  we  have 

.._  P 


1  +  sin  f  cos  V 

The  angle  <p  is  called  the  angle  of  eccentricity. 
Again,  since p=^a{\  —  c)=^a  cos^  f,  we  have 


r  = 


a  cos'  <p 


1  +  sin  ^  cos  V 


(35) 


It  is  evident,  from  this  equation,  that  the  maximum  value  of  r  in  an 
elliptic  orbit  corresponds  to  v  =  180°,  and  that  the  minimum  value 
of  r  corresponds  to  t?  =  0.  It  therefore  increases  from  the  perihelion 
to  the  aphelion,  and  then  decreases  as  the  planet  approaches  the  peri- 
helion. 


52 


THEOEETICAL   ASTROXOMY. 


i:|l 


In  tlie  case  of  the  parabola,  (p  =^  90°,  and  sin  ^  =^  c  =-  ^ ;  conse- 
quently, 

p 

1  -|-  cos  v 

But,  since  1  +  cos  v  =  2  ros'  ^v,  if  we  put  q  -=  ^p,  we  shall  lia/e 


1 

t  OS^  it) 


(36) 


in  which  q  is  the  perihrllon  distance.  In  this  case,  therefoio,  when 
V  =  ±  180°,  r  will  be  infinite,  and  the  comet  will  never  return,  but 
course  its  way  to  other  systems. 

The  angle  <f  cannot  be  applied  to  the  case  of  the  hyperbola,  since 
in  a  hyperbolic  oi'bit  c  is  greater  than  1 ;  and,  therefore,  the  eccen- 
tricity cannot  be  expressed  by  the  sine  of  an  arc.  If,  however,  we 
designate  by  i|/  the  angle  which  the  asymptote  to  the  hyperbola  makes 
with  the  transverse  axis,  we  shall  have 

e  cos  4-  =  1. 

Introducing  this  value  of  e  into  the  polar  equation  of  the  hyperbola, 

it  becomes 

^ p  cos  4^ 

cost'  +  cos  4/' 

But,  since  cos  t)  +  cos  4*  —  2  cos  J (v  +  i^)  cos  |(u  —  '4^),  this  gives 


r  = 


p  cos  4 


2  CCS  li  {v  +  4)  cos  A  (,y  —  4)" 


(37) 


It  appears  from  this  formula  that  r  increases  with  v,  and  becomes  in- 
finite when  1  -\-  e  cosi?  —  0,  or  cos  v  =  —  cos  \//,  in  which  case  v  =  180° 
—  yl/i  consequently,  the  maximum  positive  value  of  v  is  represented 
by  180°  —  'v//,  and  the  maximum  negative  value  by  -(180°  —  4). 
Further,  it  is  evident  that  the  orbit  will  be  that  branch  of  the  hyper- 
bola which  corresponds  to  the  focus  in  which  the  sun  is  placed,  since, 
under  the  operation  of  an  attractive  force,  the  path  of  the  body  must 
be  concave  toward  the  centre  of  attraction.  A  body  subject  to  a 
force  of  rej)ulsion  of  the  same  intensity,  and  varying  ae(?ording  to 
the  same  law,  would  describe  the  othci-  branch  of  the  curve. 

The  problem  of  finding  the  position  of  a  heavenly  body  as  seen 
from  any  point  of  reference,  consists  of  two  parts:  first,  the  deter- 
mination of  the  place  of  the  body  in  its  orbit;  and  then,  by  means 
of  this  and  of  the  elements  which  fix  the  position  of  the  plane  of  the 


orbit,  and 
the  po:;itio 
In  deri\ 
orbit,  we  v 
mencing  w 

20.  Sine 
+  ae,  we  n 


This  auxili 
metrical  sij 

true  anoraj 

(27)  and  w 
passage,  an 
is  to  be  con 


But 


iti/l 


The  quantit 
from  the  pei 
orbit  whose 
tlie  itiean  an 
fore,  have 


When  th( 
moan  anouu 
thr(>e  of  the 
tlioy  are  ei(^li 
helion,  provi 
helion  to  thr 
Tlie  same  n 
wo  regard,  ii 

As  soon  as 
Motion  and  i 


(37) 
los  in- 

out('<l 

^)- 
lyper- 

must 
to  a 


ing  to 


s  seen 
detcr- 
nicans 
of  tbo 


PLACE   IN  THE  ORniT. 


53 


orbit,  and  that  of  the  orbit  in  its  own  plane,  the  determination  of 
the  po:;ition  in  space. 

In  deriving  the  formula;  for  finding  the  place  of  the  body  in  its 
orbit,  we  will  consider  each  species  of  conic  section  separately,  com- 
mencinsr  with  the  ellipse. 

20.  Since  the  value  of  a  —  r  can  never  exceed  the  limits  —  ar  and 
+  ae,  we  may  introduce  an  auxiliary  angle  such  that  we  shall  ^ave 

a  —  r 

=  cos  L. 


ae 


This  auxiliary  angle  E  is  called  the  eccentric  anomahf ;  and  its  geo- 
metrical signification  may  be  easily  known  from  its  relation  to  the 

(t  —  V 
true  anomaly.     Introducing  this  value  of into  the  equation 

(27)  and  writing  t  —  T  in  place  of  t^^,  T  being  the  time  of  perihelion 
passage,  and  t  the  Knie  for  which  the  place  of  the  planet  in  its  orbit 
is  to  be  computeil,  we  obtain 


kVl  + 


m 


(t  —  T)=:E  —  esinE. 


(38) 


a 


But 


kVl  + 


m 


a 


men  I  (I'lUt/  motion  of  th ;  planet  -=  /i;  therefore 
,i{t-~  T)  =  E-'esinE. 


The  quantity  fi(t  —  T)  represents  what  would  be  the  angular  distance 
from  the  perihelion  if  the  planet  had  moved  uniformly  in  a  circular 
orbit  whose  radius  is  ft,  its  mean  distance  from  the  sun.  It  is  called 
tlie  mean  anomaly,  and  is  usually  designated  by  M,  We  shall,  there- 
fore, have 

M=-E  —  e^\nE.  (39) 

When  the  planet  or  comet  is  in  its  ])orihelion,  the  true  anomaly, 
mean  anomaly,  and  eccentric  anomaly  are  each  equal  to  zero.  Mi 
throe  of  these  increase  from  tijo  peri iiel ion  to  the  aphelion,  wliere 
tlioy  are  c^ch  equal  to  180°,  and  decrease  from  tlie  aphelion  to  the  peri- 
helion, provided  that  they  are  considered  neg;itive.  From  the  peri- 
helion to  the  aphelion  v  is  greater  than  E,  and  7v  is  great<;r  than  J/. 
The  same  relation  holds  true  from  the  aphelion  to  the  perihelion,  if 
wo  regard,  in  this  case,  tiie  values  of  t',  E,  and  3/ as  negative. 

As  soon  as  the  auxiliiin'  angle  E  is  obtained  by  means  of  the  mean 
motion  and  eccentricity,  the  values  of  r  and  v  may  be  derived.     For 


54 


THEORETICAL   ASTRONOMY. 


this  purpose  there  are  various  formulae  which   may  be  applied  in 
practice,  and  which  we  will  now  develop. 
The  equation 

a  —  r  „ 

■■  cos  L, 


gives 

This  also  gives 


or 


ae 


r  =  a(l  —  e  cosE). 


a  —  r 


P 


ae  =  a  cos  E  —  ae, 


a  cos  E —  ae, 


(40) 


which,  by  means  of  equation  (25),  reduces  to 

r  cosv  =  a  cosE  —  ae. 


(41) 


If  we  square  both  members  of  equations  (40)  and  (41),  and  subtract 
the  latter  result  from  the  former,  we  get 


}•'  sin^  i;  z=  a^  (1  —  e")  sin'  E, 


or 


r  sin  V  ==:::  a|/l  —  e"  sin  E=b  sin  E. 


(42) 


By  means  of  the  equations  (41)  and  (42)  it  may  be  easily  shown 
that  the  auxiliary  angle  E,  or  eccentric  anomaly,  is  the  angle  at  tho 
centre  of  the  ellipse  between  the  semi-transverse  axis,  and  a  line 
drawn  from  the  centre  to  the  point  where  the  prolongation  of  thu 
ordinate  perpendicular  to  this  axis,  and  drawn  through  the  place  of 
the  body,  meets  the  circumference  of  the  circumscribed  circle. 

Equations  (40)  and  (41)  give 

r(l  q^  cos  i')  =  a(l  ±  e)  (1  =P  cos  E). 

By  using  first  the  upper  sign,  and  then  the  lower  sign,  we  obtain,  by 
reduction, 

Vr  sin  ^v  =  Va{l  +  e)  sin  ^E, 

Vr  cos  A-y  =  V'a{l  —  e)  cos  l^E,  (43 ) 

which  are  convenient  for  the  calculation  of  r  and  v,  and  especially  so 
when  several  places  are  required.     By  division,  these  equations  give 


tan 


tan  ^E. 


(44) 


PLACE   IN  THE   ORBIT. 

Since  e  =  sin  <p,  Ave  have 

1  —  e      1  —  sin  <p 


55 


1  +  e       1  -{-  sin  <p 


=-tan'(45°  — ^f). 


Consequently, 
Again, 


tan  A-B=  tan  (45°  —  Af)  tan  4^?, 

V^l  +  e  =  V^l  +  sin  yi  =  l/l  -j-  2  sin  ^^  cos  l<f, 
which  may  be  written 


(45) 


or 


l/l  +  e  =  l/sin'  ^fj  -j-  COS''  A^  +  2  sin  Ay  cos  ^y, 

\/\  J^  e  =  SMX  l<f  -\-  cos  ^f . 
In  a  similar  manner  we  find 

Vl  —  e  =  —  sin  If  -f  cos  ^p. 
Fi'om  these  two  equations  we  obtain 


l/l  +  e  +  V'l  —  e  =  2  cos  Av^ 
l/l  +e  -  l/l  —  t'=^  2  sin  i^^, 


(46) 


which  are  convenient  in  many  transformations  of  equations  involving 
c  or  ip. 
Equation  (42)  gives 

.     „      7'  sin  V  p  sin  v 

6  6  (1  +  e  cos  v) 

but  p'^a  cos^  ^,  ant!  b^=a  cos  ^,  hence 

r  sin  V         cos  f  sin  v 


smE- 


acosf       1  -j-  e  cosv 


(47) 


Equation  (41)  gives 


coaE: 


r  cos  V  -{-  ae 
a 


p  cosv 


or 


cosjEJ: 


a  (1  -|-  e  cos  1') 
jo  cos  V  -\-  ae  -j-  w  ■  cos  v 


+  e, 


a{l  -\-  e  cos  I'j 

and,  putting  a  cos^  ^  instead  of  p,  and  sin  (f  for  c,  we  get 

_,        cos  V  4-e 
cosE  = 


(48) 


1  -{-  €  COS  V 

If  we   multiply  the  first  of  equations  (43)  by  cos^E,  and  the 


m 


THEORETICAL  ASTEONOMY. 


second  by  sin^E,  successively  add  and  subtract  the  products,  and 
reduce  by  means  of  the  preceding  equations,  we  obtain 


lin  ^(v-{-  E)  —  -J-  cos  If  sin  E, 
■in i (v  —  £)  —  a/-  sin  l<p  sin E. 


(49) 


The  perihelion  distance,  in  an  elliptic  orbit,  is  given  by  the  equa- 
tion 

5  =:  a  (1  —  e). 

21.  The  difference  between  the  true  and  the  mean  anomaly,  or 
V  —  M,  is  called  the  equation  of  the  centre,  and  is  positive  from  the 
perihelion  to  the  aphelion,  and  negative  from  the  aphelion  to  the 
perihelion.  When  the  body  is  in  either  apsis,  the  equation  of  the 
centre  will  be  equal  to  zero. 

We  have,  from  equation  (39), 

E=M-{-  e  sin E. 
Expanding  this  by  Lagrange's  theorem,  we  get 

FiE)  =  F{M)  +  sm  M^^^    •  ^  +  -^^^(  sm-  M  ^^  )  ^ 


sin'3/— 
Let  us  now  take,  equation  (40), 


(50) 


FiE)  =  (.l-e  cos E)    =^, 

and,  consequently, 

F(M)  =  (l  —  e  cos  3rf\ 

Therefore  we  shall  have 

^  ^  (1  _  e  cos  il/)"  —  2e'  sin'  J/(l  —  e  cos  M)~' 

—  e»  -p^  (sin'  il/(l  —  e  cos  M)    )  — 

Expanding  these  terms,  and  performing  the  operations  indicated,  we 
get 

^  =  1  +  2e  cos  if  +  ^  (6  cos*  M—A  sin'  M) 


+  I  (16  cos"  M~  36  sin'  M  cos  ilf )  +  . . . , 


PLACE  IN  THE  ORBIT. 


67 


wliich  reduces  to 


^=^l+2e co9i>/+|(l+5 CG2 2il/)+| (13 cos33/+3 cosiV)+ ....    (51) 
Equation  (22)  gives 


dv  =  ^ 


2fdt 


and,  siuce/=p]/p(l  +  m),  we  have 


r 


(52) 


or 


rf„=^Ki+j!^«Vr=7'd^. 


a* 


But 5 =  n,  and  therefore 


a 


dv  ^^^l  —  e'-,  iidt  =  Vl  —  e'  -,  dM. 


By  expanding  the  factor  ^/l  —  e^,  we  obtain 

T/r^=^=-=  1  —  V  — ^e*— . . . , 


and  hence 


dv  =  (\~  le" 


.)  -,  dM. 


a 


Substituting  for  —  its  value  from  equation  (51),  and  integrating,  we 
get,  since  w  =^  0  when  M=^  0, 

V— l/=2e  sin  iJf+^6' sin  2J/+^  (13  sin  3ilf— 3  sin  J/)  +     ,     (53) 

Mhieh  is  the  expression  for  the  equation  of  the  centre  to  terms  involving 
c^.  In  the  same  manner,  this  series  may  be  extended  to  higher  powers 
of  c 

When  the  eccentricity  is  very  small,  this  series  converges  very 
rapidly;  and  the  value  of  v  —  If  for  any  planet  may  be  arranged  in 
a  table  with  the  argument  M. 

For  the  .purpose,  however,  of  com[)uting  the  places  of  a  heavenly 
body  from  the  elements  of  its  oi'bit,  it  is  proCorable  to  solve  the 
equations  which  give  v  and  J5^ directly;  and  when  the  eccentricity  is 


58 


THEORETICAL   ASTRONOMY. 


lif' 


very  great,  this  mode  is  indispensable,  since  the  series  will  not  in 
that  case  be  sufficiently  convergent. 

It  will  be  observed  that  the  formula  which  must  be  used  in  obtain- 
ing the  eccentric  anomaly  fi'om  the  mean  anomaly  is  transcendental, 
and  hence  it  can  only  be  solved  either  by  series  or  by  trial.  But 
fortunately,  indeed,  it  so  happens  that  the  circumstances  of  the  celes- 
tial motions  render  these  approximations  very  rapid,  the  orbits  being 
usually  either  nearly  circul.!".  or  else  very  eccentric. 

If,  in  equation  (50),  we  put  F{E)  =  E,  and  consequently  F{M) 
=  31,  Ave  shall  have,  performing  the  operations  indicated  and  reducing, 

^  :=  J/  +  e  sin  J/  +  .Je^  sin  231  +  &c.  (54) 

Let  us  now  denote  the  approximate  value  of  E  computed  from  this 
equation  by  Eq,  then  will 

E,+  £.E,=^E, 

in  which  a£^,  is  the  correction  to  be  applied  to  the  assumed  value  of  E. 
Substituting  this  in  equation  (39),  we  get 

3I==  E^-{-  ^E^  —  esinE^  —  e  cos  ^'o^-Eo ; 

and,  denoting  by  J/„  the  value  of  31  corresponding  to  j%,  we  shall 

also  have 

3f^=E^  —  e  sin  Eg. 

Subtracting  this  equation  from  the  preceding  one,  we  obtain 


ilf  —  3fo 
—  e  cos  En 


A^.. 


m 


It  remains,  therefore,  only  to  add  the  value  of  i^Eg  found  from  this 
formula  to  the  first  assumed  value  of  E,  or  to  £"„,  and  then,  using 
this  for  a  new  value  of  E^^,  to  proceed  in  precisely  the  same  manner 
for  a  second  approximation,  and  so  on,  until  the  correct  value  of  E  is 
obtained.  When  the  values  of  E  for  a  succession  of  dates,  at  equal 
intervals,  are  to  be  computed,  the  assumed  values  of  E^  may  be  ob- 
tained so  closely  by  interpolation  that  the  first  approximation,  in  the 
manner  just  explained,  will  give  the  correct  value;  and  in  nearly 
every  case  two  or  three  approximations  in  this  manner  will  suffice. 

Having  thus  obtained  the  value  of  E  corresponding  to  31  for  any 
instant  of  time,  we  may  readily  deduce  from  it,  by  th»  formuhe 
already  investigated,  the  corresponding  values  of  r  and  v. 

In  the  case  of  an  ellipse  of  very  great  eccentricity,  corresponding 
to  the  orbits  of  many  of  the  comets,  the  most  convenient  method  of 


com])ut 
manner 
sider  lie 
for  dete 
lor  el  lip 
and  e  = 


PLACE   IN  THE  ORBIT. 


59 


conipnting  r  and  v,  for  'my  instant,  is  somewhat  difforont.  The 
manner  of  proceeding  in  the  computation  in  such  cases  we  shall  con- 
sider hereafter;  and  we  will  now  proceed  to  investigate  the  formuhc 
for  determining  r  and  v,  when  the  orbit  is  a  parabola,  the  formulte 
lor  elliptic  motion  not  being  applicable,  since,  in  the  parabola,  a  =^  gc  , 
and  c  =  1. 

22.  Observation  shows  that  the  masses  of  the  comets  are  insensible 
in  comparison  with  that  of  the  sun;  and,  consequently,  in  this  case, 
m  =  0  and  equation  (52),  putting  for  p  its  value  2q,  bef'omes 


or 


kV2q  dt  =  ?-Mi', 
^  cos*  .Vy 


which  may  be  written 
hdt 


V2f 


1(1  +  tan''  y)  see'  ^vdv  =  (14-  tan'  iv)  d  tan  Av. 


Integrating  this  expression  between  the  limits  T  and  t,  we  obtain 

k(t-T) 


V2 


=;  tan  2^+3  tan'  Iv, 


(55) 


wliich  is  the  expression  for  the  relation  between  the  true  anomaly 
and  the  time  from  the  perihelion,  in  a  parabolic  orbit. 

liCt  us  now  represent  by  Tq  the  time  of  describing  the  arc  of  a 
parabola  corresponding  to  v  =  90° ;  then  we  shall  have 


V2n' 


4 
3' 


or 


l/2~~^o 


Sk 


Now,  — >■-  is  constant,  and  its  logarithm  is  8.5621876983;  and  if  we 

I  take  7  =  1,  which  is  equivalent  to  supposing  the  comet  to  move  in 
a  ^)arabola  whose  perihelion  distance  is  equal  to  the  semi-transverse 
axis  of  the  earth's  orbit,  we  find 


log- 


dayi 


2.03987229,  or  r^  =  109.61558  days ; 


that  is,  a  comet   moving  in  a  parabola  whose   perihelion  distance 


^^mmmmmmmm 


60 


THEORETICAL   ASTKONOMY. 


is  equal  to  the  mean  distance  of  the  earth  from  tlie  sun,  requires 
109.61558  days  to  describe  an  arc  corresponding  to  v  ^^  90°. 

E(juation  (55)  contains  only  such  quantities  as  are  comparable  with 
each  other,  and  by  it  t~  T,  the  time  from  the  perihelion,  may  be 
readily  found  when  the  remaining  terms  are  known;  but,  in  order 
to  find  V  from  this  formula,  it  will  be  necessary  to  solve  the  equation 
of  the  third  degree,  tan  Ji?  being  the  unknown  quantity.  If  we  put 
X  =  tan  it',  this  equation  becomes 

3?  -\-  Sx  —  a  =  0, 

in  which  a  is  the  known  quantity,  and  is  negative  before,  and  positive 
after,  the  perihelion  passage.  According  to  the  general  pri  iciple  in 
the  theory  of  equations  that  in  every  equation,  whether  con  plete  or 
incom})lete,  the  number  of  positive  I'oots  cannot  exceed  the  number 
of  variations  of  sign,  and  that  the  number  of  negative  roots  cannot 
exceed  the  number  of  variations  of  sign,  when  the  signs  of  the  terms 
containing  the  odd  powers  of  the  unknown  quantity  are  changed,  it 
follows  that  when  a  is  positive,  there  is  one  positive  root  and  no 
negative  root.  When  a  is  negative,  there  is  one  negative  root  and 
no  positive  root;  and  hence  we  conclude  that  equation  (55)  can  have 
but  one  real  root. 

We  may  dispense  with  the  direct  solution  of  this  equation  by 
forming  a  table  of  the  values  of  v  corresponding  to  those  of  t  —  T 
in  a  parabola  whose  perihelion  distance  is  equal  to  the  mean  distance 
of  the  earth  from  the  sun.  This  table  will  give  the  time  correspond- 
ing to  the  anomaly  v  in  any  parabola,  whose  perihelion  distance  is 

q,  by  multiplying  by  q'^,  the  time  which  corresponds  to  the  same 
anomaly  in  the  table.     We  shall  have  the  anomaly  v  corresponding 

3 

to  the  time  t  —  ^T by  dividing  t  —  Thy  q^,  and  seeking  in  the  table 
the  anomaly  corresponding  to  the  time  resulting  from  this  division. 

A  more  convenient  method,  however,  of  finding  the  true  anomaly 
from  the  time,  and  the  reverse,  is  to  use  a  table  of  the  form  gene- 
rally known  as  Barker's  Table.  The  following  will  explain  its  con- 
struction : — 

Multiplying  equation  (55)  by  75,  we  obtain 


(t—T)  =  75  tan  ^v  +  25  tan»  ^v. 


Let  us  now  put 


M=^  75  tan  J,v  +  25  tan«  Av, 


PLACE   IN   THE  ORBIT. 


61 


aiul  CI 


'  V2 


,  which  is  a  constant  qnantity;  then  ■will 


The  value  of  Q  is 
Again,  let  us  take 


^(t—T)  =  M. 


log  C„  =  9.9601277069. 
C 
2* 


which  is  called  the  mean  daily  motion  in  the  parabola ;  then  will 

M=  m  (t  —  T)  =  75  tan  ^v  +  25  tun'^v. 

If  Ave  now  compute  the  values  of  31  corresponding  to  successive 
values  of  v  from  r  =  0°  to  v  =  180°,  and  arrange  them  in  a  table 
witli  the  argument  v,  we  may  derive  at  once,  from  this  table,  for  the 
time  {t  —  T)  either  3/ when  v  is  known,  or  v  when  3I=^m  {t  —  T) 
is  known.  It  may  also  be  observed  that  when  t  —  T  is  negative,  the 
value  of  V  is  considered  as  being  negative,  and  hence  it  is  not  neces- 
sary to  pay  any  further  attention  to  the  algebraic  sign  of  t  —  T  than 
to  give  the  same  sign  to  the  value  of  i'  obtained  from  the  table. 

Table  VI.  gives  the  values  of  3/ for  values  of  v  from  0°  to  180°, 
with  ditfcrences  for  interpolation,  the  application  of  which  will  be 
easily  understood. 

23.  When  v  approaches  near  to  180°,  this  table  will  be  extremely 
inconvenient,  since,  in  this  case,  the  differences  between  the  values  of 
M  for  a  difference  of  one  minute  in  the  value  of  v  increase  very 
rapidly ;  and  it  will  be  very  troublesome  to  obtain  the  value  of  v 
from  the  table  with  the  requisite  degree  of  accuracy.  To  obviate 
the  necessity  of  extending  this  table,  we  proceed  in  the  following 
manner: — 

Equation  (55)  may  be  written 

k(t—T) 


1/2  «t 


=  ]tan':Vy  (1  +3cot'U'); 


and,  multiplying  and  dividing  the  second  member  by  (1  +  cot^  Jr)^ 
wo  shall  have 


kit—T) 
V2q^ 


itan.A.(l+cot4.)'Ji±:|^. 


62 


THEORETICAL   ASTRONOMY. 


But  1  +  cot"  \V  ^ 


Sill  V  tan  \v 
k  it  —  T)  ^ 


and  consequently 

8         1  +  8  cot'  ^t' 
Bsiu'v*  (.l-j-cot';iy)''" 


Now,  when  v  approaches  near  to  180°,  cot  Av  will  be  very  small,  .iiid 
the  second  factor  of  the  second  member  of  tliis  equation  will  nearly 
=  1.  Let  us  therefore  denote  by  w  the  value  of  v  on  the  supposition 
that  this  factor  is  equal  to  unity,  which  will  be  strictly  true  when 
v  =  180°,  and  we  shall  have,  for  the  correct  value  of  v,  the  Ibllowing 
equation : 

Ay  being  a  very  small  quantity.     We  shall  therefore  have 
8 


sin' to 


=n  3  tan  j  (io  +  \)  +  tan'  i  {w  +  A„), 


and,  putting  tan  |w  =  d,  and  tan  ^  a^  —  x,  we  get,  from  this  equation, 

tf'        ~~     l  —  Ox'^  0-  —  Oxf 

Multiplying  this  through  by  ^■' (1  —  ^.i;)^  expanding  and  reducing, 
there  results  the  following  equation : 

1  +  3tf'  =  SO  (1  +  40'  -\-20*^0»)x  —  SO^  (1  +  AO"  +  2fl*  +  tf«)  a^ 
4-  ^^^  (2  +  QO'  +  80*  4-  fl")  3^. 

Dividing  through  by  the  coefficient  of  x,  we  obtain 


_    1  +  3fl* 

3^tf(r+4^'  +  2tf*-f  fl«) 

Let  us  now  put 


:  X  —  Ox"  +  -r 


,   ,   (P{%  +  &0''-\-2,0'-\-(!^)a? 


3  (1  +  4<y^  +  2ff*  ^-  tf«) 


l  +  3fl^ 


30  (1  +  4tf'  +  2tf*  +  (fi) 


=  2/; 


then,  substituting  this  in  the  preceding  equation,  inverting  the  series 
and  reducing,  we  obtain  finally 

,   „  ,   ,  <?H4  +  180'  +  9<?*  +  5^)  .  ,    „ 

But  tan  jAfl  =  X,  therefore 

Ao  =  2a;  — 3«»  + 


PLACE   IN   THE   OllIJIT. 


63 


Substituting  in  this  the  value  of  x  above  found,  and  reducing,  we 
obtiiin 


\  =  2y  +  20y'  + 


3  (1  +  40*  +  2tf*  +  (fi) 


if  +  &c. 


For  all  the  cases  in  which  this  c<iuation  is  to  be  applied,  the  third 
term  of  the  second  member  will  be  insensible,  and  we  shall  have,  to 
a  siutFieient  degree  of  approximation, 

^,  =  2!/-\-20,f. 

Tabic  VII.  gives  the  values  of  a^,  expressed  in  seconds  of  arc, 
corresponding  to  consecutive  values  of  w  from  lo  =-  155°  to  iv  -- 180°. 
In  the  application  of  this  tivblc,  we  have  only  to  compute  the  value 
of  ^f  precisely  as  for  the  ease  in  which  Table  VI.  is  to  be  used, 

namely, 

M=m{t  —  T): 

then  will  to  be  given  by  the  formula 

'1200 

sin  W—-\  -zrfT, 

since  we  have  already  found 

k  (t  -~T)_      8 
]/2o^         Ssin'w;' 

or 


siniu 


__'!    8qW2      _  '[200 


Having  computed  the  value  of  w  from  this  equation.  Table  VII. 
will  furnish  the  corresponding  value  of  AqJ  and  then  we  shall  have, 
for  the  correct  value  of  the  true  anomaly, 

t>  =  ii;  +  Ao, 

which  will  be  precisely  the  same  as  that  obtained  directly  from  Table 
VI.,  when  the  second  and  higher  orders  of  differences  are  taken  into 
account. 

If  V  is  given  and  the  time  t  —  Tis  required,  the  table  Avill  give, 
by  inspection,  an  approximate  value  of  A,  using  v  as  argument,  and 
then  IV  is  given  by 


64 


THEORETICAL   AHTROXOMY. 


The  exact  valuo  of  a„  is  tlion  foiiiid  irom  the  table,  and  hence  \vc 
derive  that  of  «' ;  and  finally  t  —  T  from 


t-T^ 


200 


J 


C'o     aWiv 


24.  The  problem  of  finding  the  time  t  —  T  when  the  true  anomaly 
is  given,  may  also  bo  solved  conveniently,  and  especially  so  when  v  is 
small,  by  the  following  process: — 

Equation  (55)  is  easily  transformed  into 

from  which  we  obtain,  since  q^=r  cos^^v. 


2/2 


_    /Hin^\_     /sinjtt\» 
~    \   1/2  /         \   1/2  / 


Let  us  now  put 
and  we  have 

Con'^cciuently, 


Uit-T) 


2r^ 


sin  X  = -rrz., 

1/2 


3  sin  a;  —  4  sin' a;  =  sin  3a;. 


t 


2    f   .    o 
3^r   sm3a,-. 


which  admits  of  an  accurate  and  convenient  numerical  solution.  To 
facilitate  the  calculation  we  put 

,,      sin  3a; 
sinv 

the  values  of  which  may  be  tabulated  with  the  argument  v.  When 
t,  =_-  0,  we  shall  have  iV=  fV  2,  and  when  v  =  90,  we  have  N=\; 
from  which  it  appears  that  the  value  of  iV  changes  slowly  for  values 
of  V  fi-om  0°  to  90°.  But  when  ?'=^180°,  we  shall  have  N=cc; 
and  hence,  when  v  exceeds  90°,  it  becomes  necessary  to  introduce  an 
auxiliary  different  from  N.     We  shall,  therefore,  put  in  this  case, 


from  win 
wlien  V  - 


and,  whei 


in  which  1 

when  V  is 
Tal)Ie  \ 

tion,  for  vi 
for  those  0 

25.  We 

from  the  ei 
formulttj  fo 
only  that  c 
tive  or  im 
auxiliary  q 
the  two,  an 
For  this 


When  V  =^ 
nominator  \ 
180°-i^  an 
for  the  max 
vanish  for  tl 
either  case,  i 
In  the  hj 
quontly,  we 

We  have,  ah 
Let  us  now  ] 


N'  =  Nsinv  =  sin  3a;; 


PLACE  IN   THE  ORniT. 


60 


from  wliich  it  appears  tlmt  N' -'-■  1  when  r      90°,  and  that  N'^^Vi 
when  V  -    180°.     Tlicrcforc  we  have,  finally,  when  v  is  less  than  DO", 


t-T. 

and,  when  v  is  greater  than  90°, 

t—T-^ 


U 


Nr  sin  v, 


2 


-  N'r 


in  which  log  -  -  =  1.. '3883272995,  from  which  t  —  3'  is  easily  derived 

when  V  is  known. 

Tai)le  VIII.  gives  the  valncs  of  N,  with  differences  for  interpola- 
tion, for  vail  (•;  of  V  from  v  =-  0°  to  v  --  90°,  and  the  values  of  N' 
for  those  of  v  from  v  :^  90°  to  v  =  180°. 

25.  We  shall  now  consider  the  ease  of  the  hyperbola,  which  differs 
from  the  ellipse  only  that  e  is  greater  than  1 ;  and,  consequently,  the 
formuhe  for  elli])tic  and  hyperbolic  motion  will  differ  from  each  other 
only  that  certain  quantities  which  are  positive  in  the  ellipse  are  nega- 
tive or  iniMginary  in  the  hyperbola.  We  may,  however,  introduce 
auxiliary  quanuiies  whicli  will  serve  to  preserve  the  analogy  between 
the  two,  and  j  c'  u>  mark  the  necessary  distinctions. 

For  this  purpose,  let  ris  I'esume  the  equation 


r  = 


p  cos  •4« 


2  cos  A  (v  -f-  ■4')  cos  ^  (y  —  4.) 


When  V  =  0,  the  factors  cos  J(i;  +  -J/)  and  cos|(y  —  ^l/)  in  the  de- 
nominator will  be  equal;  and  since  the  limits  of  the  values  of  v  are 
180°— '4/  and  —(180°  —  -v/.),  it  follows  that  the  first  factor  will  vanish 
for  the  maximum  positive  value  of  r,  and  that  the  second  factor  will 
vanish  for  the  maximum  negative  value  of  v,  and,  therefore,  that,  in 
cither  case,  /•  =  00. 

In  the  hyperbola,  the  semi-transverse  axis  is  negative,  and,  conse- 
quently, we  have,  in  this  case, 

^j  =  a(e' — 1),  or  a  =jj  cot'^-. 

We  have,  also,  for  the  perihelion  distance, 

q  =  a(e  —  1). 
Let  us  now  put 


taniF 


=  tan  -^vy 


e  — 1 


(56) 


:l 


ee 


THEORETICAL   ASTRONOMY, 


which  is  analogous  to  the  formula  for  tlic  eccentric  anomaly  ^  in  an 

ellipse :  and,  since  c  —  — -  .  we  shall  have 
*     '         '  cos  4. 


and,  consequeiitly, 


e  —  1       1  — C0S4      ,     ., , , 

— -~-  =  — =  tan'  ^4, 

e  -j-  1       1  -j-  cos  4 


tan  A-F  =  tan  ^ V  tan  A4. 
We  shall  now  introduce  an  auxiliary  quantity  a,  such  that 

1  -f  tin  hF 


(57) 


ff=:Un(Ao°  +  }<F) 


whence  we  derive 
and  also 


tan  A  J'---      ,  ^ 


cos  2  ( I'  —  4) 

COS  I  iv  -f-  ^) 


1  —  tan^i'^' 
ff  —  1 


(58) 


(59) 


This  last  equation  shows  that  ff  ==  1  when  the  comet  is  in  its  perihe- 
lion; <7  =  00  when  «-=180° — 4'-,  and  <t=:  0  when  v  =  ~  (180=  —  i). 


Since  tan  F^ -r^rfT^'  ^ve  shall  have 

1  — tau'ii^ 


tan 


Squaring  this  equation,  adding  1  to  both  members,  and  reducing  we 
obtain 

-l^==d<r  +  ^\.  (61) 

lieplacing  a  in  this  equation  by  its  value  from  equation  (59),  we  get 
1  cos'  I  (v  +  4)  +  cos'  Ky  —4) 


or 


cosjF  ^"'  2  cosf('(;"+ 4)cosJ(.'y  ~-  ;■)  ' 
1  +  cos  V  cos  4  (e  +  cos  v)  cos  4 


cos  F  ~'  2  oosT(t;  -\-  4)  cos  -^  (y  ---4)   "   2  cos'A  (v  +  4)  cos  ^  (v  -  4 
which  reduces  to 


1      r  (<3  -f  cos  v) 

cos  /'■'  ~         p 


(62) 


IP  wo  add 


Taking  fir 
we  get 


These  cqua 
to  tliose  j»r( 
give,  by  dii 


which  is  id 

verify  the  o 

Multiplyi 

e^  —  1  its  va 


(60)    I  Further,  we 


which,  comb 


If  we  squarp 
getlior,  reduci 


^yp  might  als 
(<j'"3);  but  sue 
desirable,  it  cii 
luive  already  ( 


PLACE   IN  THE  ORBIT. 


67 


If  wc  add  :4:  1  to  both  members  of  this  equation,  we  shall  have 

1  ~-  cos  F r  f  (!  =p  1)  n.  qi  cos  v) 

cos/'  j) 

Taking  first  the  upper  sign,  and  then  the  lower  sign,  and  reducing, 
we  get 


!/• 


r  sm  iv  = 


Vafe-irl) 
VcofiF 


sm\F, 


,/-        ,         Vaie  —  1) 

V  r  cos  hv  =; ^_--_-__  (,03  ii- . 

l/cos  F 


(63) 


Tlieso  equations  for  finding  i  and  r,  it  will  be  observed,  are  analogous 
to  those  previously  investigated  for  an  elliptic  orbit.  These  equations 
give,  by  division, 


tan  J,v  —  \/''  —  -  tan  \F, 


which  is  identical  with  the  equation  (56),  and  may  be  employed  to 
verify  the  computation  of  ;•  and  v. 

Multiplying  the  last  of  equations  (63)  by  the  first,  putting  for 
r  —  1  its  value  tan'"  \//,  and  reducing,  we  obtain 


r  sin  V  —  u  tan  ^  tan  jP— -  ^a  tan  4 1  <r  —  -  1. 

Further,  we  have 

p  cos  V  ar  (e  4-  cos  v) 

r  cos  V  =  :,  -  ,  -    —  =  oe ^ , 

1  -{-  e  cos  V  p 

wliieh,  combined  with  equation  (62),  gives 

r  cos v~-  %[  e — —rr  I -•=  A« I  2c  —  a  —  -  1  • 

\         cosi  /       *•    \  <tJ 


(6-t) 


(65) 


If  wc  square  these  valios  of  r  s'mv  and  r  cosr,  add  the  results  to- 
gether, reduce,  and  ex*:fact  the  square  root,  we  find 


Ach-')=H'i'+l)-'\ 


(66) 


^Vo  might  also  introduce  the  auxiliary  quantity  er  into  the  equations 
(O.'j);  but  such  a  transformation  is  hardly  necessary,  and,  if  at  all 
desirable,  it  can  be  easily  effected  by  means  of  the  formidte  which  we 
have  already  derived. 


68  THEORETICAL   ASTRONOMY. 

26.  Let  us  now  resume  the  equatioa 

cos  ;\  {v  —  4') 

corf  A  (y  +  4)' 

Diiferentiiiting  this,  regarding  t^  as  constant,  we  have 

sln^/ 


dff 


■dv, 


2  cos^  A  i,v  +  4) 
and,  divi.iing  this  equation  by  the  preceding  one,  we  get 


sin  4 


cdv. 


But 

consequently, 
which  gives 


rfff 

ff        2  cos  ^  iv  +  4)  cos  h  {v  —  4) 

p  cos  4 

2  cos  ^  (v  -\-  4)  cos  ;^  {v  —  4)' 

rfff       r  tan  4 


r'dv 


P 


pr 


a  tan  4 


dv, 


d(T. 


Substituting  this  value  of  ')^dv  in  equation  (22),  and  putting  instead 
of  2/  its  value  kVp,  from  equation  (30),  the  mass  being  considered  as 
insensible  in  comparison  with  that  of  the  sun,  we  get 

kVp  dt  =  —i — -  dff. 
•*  a  tan  4 

Then,  substituting  for  r  its  value  from  equation  (66),  and  for  p  its 
value  a  tan^  ^,  we  have 

kVp  dt  =  a'  tan  4  Ue  f  1  +  ^  j  —  -  W<r, 

Integratiug  this  between  the  limits  T  and  t,  Ave  obtain 

kVp(t— T)==  aHan^ I  },e(^<T  —  -) —log, A,  (67) 

in  which  log,  a  is  the  Naperian  or  hyperbolic  logarithm  of  a.  Since 
l/p  =  y  a  tan  ;//,  if  we  put 


in  wliic] 


in  whicl 
shall  lia' 


If  Ave  m 
the  modt 


M-c  shall 


wherein  1 
liot  us 
have 


and  also 
Tlierefore 


This  equal 
when  a,  e. 
sohition  o 
and  radius 
sncocssive 
If  we  d 
able,  we  gc 


Honce,  if  \ 
I't'sponding 
of  F  may  I 


a* 


PLACE   IN   THE  ORBIT.  69 

iu  wliich  V  is  the  mean  daily  motion ;  and  if  we  also  put 

in  which  Nq  corresiionds  to  the  nuean  anomaly  JI  in  an  ellipse,  we 
shall  have,  from  equation  (67), 


No  =^  Uiff  —  -j  —  log,  a. 


(68) 


If  we  multiply  both  members  of  this  equation  by  A  =  0.434294482, 
the  modulus  of  the  common  system  of  logarithms,  and  put 


we  shall  have 


a' 
iV==--^cA|(T  — -\  — logff, 


wherein  log  A  =  9.6377843113,  and  log^Z;  -=  7.8733657527. 

Let  us  now  introduce  i^^into  this  formula;  and  for  this  purpose  we 
have 


tanF=^, 


log  ff  =  log  tan  (45°  +  ^F). 


(69) 


and  also 

Therefoi'e  we  obtain 

iY^eA  tani^—  log  tan  (45°  +  ^F). 

This  eqiuition  will  give,  directly,  the  time  t  —  Tfrom  the  perihelion, 
when  a,  e,  and  i'^aro  known;  but,  since  it  is  tvanscendental,  in  the 
solution  of  the  inverse  problem,  that  of  finding  the  true  anomaly 
and  radius-vector  from  the  time,  the  value  of  F  can  only  be  found  by 
successive  api)roximations. 
If  we  differentiate  the  last  equation,  regarding  -AT and  F  as  vari- 


able, we  get 


dN  =---  -Ari^  (e  —  cos  F)  dF. 


Hence,  if  we  denote  an  approximate  value  of  F  by  F„  and  the  cor- 
responding value  of  N  by  N„  the  correction  ajP,  to  the  assumed  value 
of  F  may  be  computed  by  the  fornuda 


^F, 


(X—N,)co»'F, 
).  (e  —  cos  F,) 


70 


TIIEORETrCAL   ASTRONOMY. 


This  correction  being  applied  to  F„  a  nearer  approximation  to  the 
true  value  of  F  will  be  obtained;  and  by  repeating  tlie  operation 
there  results  a  still  closer  approximation.  This  process  may  be  con- 
tinued until  the  exact  value  of  F  is  found,  and,  when  several  suc- 
cessive places  are  required,  the  first  assumed  value  may  be  estimated, 
in  advance,  so  closely  that  a  very  few  trials  will  suffice.  In  practice, 
however,  cases  will  rarely  occur  in  which  this  formula  will  be  applied, 
since  the  probability  of  liyperboHc  motion  is  small,  and,  whenever 
any  positive  indication  of  an  eccentricity  greater  than  1  has  been 
found  to  exist,  it  has  only  been  after  a  very  accurate  series  of  observa- 
tions has  been  introduced  as  the  basis  of  the  calculation.  For  a 
majority  of  the  cases  which  do  really  occur,  the  most  accurate  and 
convenient  method  of  finding  r  and  v  will  be  explained  hereafter. 

27.  If  we  consider  the  equation 

M=  E — esiuJEJ, 

we  shall  see  that,  when  logarithms  of  six  or  seven  decimals  are  used, 
the  error  which  may  exist  in  the  determination  of  E  when  M  and  e 
are  given,  will  increase  as  e  increases,  but  in  a  much  greater  ratio; 
and,  when  the  eccentricity  becomes  nearly  equal  to  that  of  the  para- 
bola, the  error  may  be  very  great.  In  the  case  of  hyperbolic  motion, 
also,  the  numerical  solution  of  equation  (69),  when  c,  —  1  is  very 
small,  and  with  the  ordinary  logarithmic  tables,  becomes  very  un- 
certain. This  can  only  be  remedied,  when  equations  (-39)  and  (G9) 
ai'o  employed,  by  using  more  extended  logarithmic  tables;  and  when 
the  orbit  diffiu's  only  in  an  extremely  slight  degree  from  a  parabola, 
even  with  the  most  extended  logarithmic  tables  which  have  been 
constructed,  the  error  may  be  very  large.  For  this  reason  we  have 
recourse  to  other  methods,  which  will  give  the  required  accuracy 
without  introducing  inconveniences  which  are  proportionally  great. 

We  shall,  therefore,  now  proceed  to  develop  the  formuhc  for  find- 
ing the  true  anomaly  in  ellipses  and  hyperbolas  which  diffin*  l)ut 
little  from  the  parabola,  such  that  they  will  furnish  the  requirid 
accuracy,  when  the  exact  solution  of  equations  (39)  or  (69)  with  the 
logarithmic  tables  in  common  use  is  impossible. 

For  this  purpose,  let  us  I'csumo  fjquation  (22),  which,  by  substi- 
tuting for  2/  its  value  h\/ 1),  the  mass  of  the  comet  being  neglectod 
in  comparison  with  that  of  the  sun,  becomes 


k  i/p  dt  =  tHi\ 


PLACE  IN  THE  ORBIT. 


71 


or 


ki/pdt=p^ 


dv 


(1  -f  e  cos  vy 
Let  us  now  put  u  =  tan  Ju,  and  we  shall  have 


C03V 


1  — tt 

!  +  «■ 


,''<i^  =  l 


2du 


Substituting  these  values  in  the  preceding   equation,  and  putting 
— —  =  I,  we  get 


(l  +  ey   U  +  'V/' 


or,  since p  =  q{l  +  e). 


ki/l-{-edt  _  (1  4-  u')  dii 
'-  (1  +  iu^r  ' 


2n'^ 


Let  us  now  develop  the  second  member  into  a  series.     This  may  be 
written  thus: 

and  developing  the  last  factor  into  a  series,  we  obtain 

(1  +  iuT'  =  1  —  2iu^  +  3iHi'  —  4Pu^  +  &c. 
Consequently, 

(1  +  u')  (1  +  iV)~'  =  1  +  «'  -  2i(ii«  +  u*)  +  SiHu'  +  u^) 

—  iPiU^  +  'U^)  +  .... 

Multiplying  this  equation  through  by  du,  and  integrating  bctv,-ecn 
the  limits  Tai^d  t,  the  result  is 


2q^ 


—  4i'  iijii'  4-  Ju")  +  &c.  (70) 


III  the  case  of  the  parabola,  c  =  1  and  i  =  0,  and  this  equation  becomes 
identical  with  (55). 
Let  us  now  put 

k(t—T)yT~-^c 


2.^ 


U+]U\ 


(71) 


72 

and  also 


THEOBETICAL   ASTRONOMY. 


l7=taniF/ 


then  the  angle  Fwill  not  be  the  true  anomaly  in  the  parabola,  but 
an  angle  derived  from  the  solution  of  a  cubic  equation  of  the  same 
form  as  that  for  finding  the  parabolic  anomaly ;  and  its  value  may 
be  found  by  means  of  Table  VI.,  if  we  use  for  M  the  value  com- 
puted from 

76k(t--T)      jTT'e 

,-r,   I      •V-2- 
1/29 


M- 


Let  U  be  expanded  into  a  scries  of  the  form 

XJ  =^U  -\-  a.i  -^  fii^  -\-  yl^  -\-  ... 


which  is  evidently  admissible,  a,  ft,  y, ....  being  functions  of  u  and 
independent  of  /.  It  remains  now  to  determine  the  values  of  the 
coefficients  a,  /?,  ;',  &c.,  and,  in  doing  so,  it  will  only  be  necessary  to 
consider  terms  of  the  third  order,  or  those  involving  P,  since,  for 
nearly  all  of  those  cases  in  which  the  eccentricity  is  such  that  terms 
of  the  order  i^  will  sensibly  affect  the  result,  the  general  formulae 
already  derived,  with  the  ordinary  means  of  solution,  will  give  tli<' 
required  accuracy.     We  shall,  therefore,  have 

U -\- iU' ==  u -\- ai -}-  ISi'  +  yi?  +  J- (u  +  tti  +  /?i»  +  yPf, 

or,  again  neglecting  terms  of  the  order  i^, 

C7  -f  J  [/'  =  tt  +  J  It'  +  i  (1  +  11')  a  +  i'  (lla^  +  (1  +  «')  /5) 
-\.i\.y-\-2ua,3+{l+U^)y). 


But  we  have  already  found,  (70), 

k(t  — 

2<f 


TlVl±^^u+iw 


u 


i«' 


Since  the  first  members  of  these  equations  are  identical,  it  follows,  by 
the  principle  of  indeterminate  coefficients,  that  the  coefficients  of  the 
like  powers  of  i  are  equal,  and  we  shall,  therefore,  have 

,,a«  +  (1  +  w«)  /?  =  +  3  (|«»  +  jn'), 
.1  a'  +  2uai3  +  (1  +  «»)  ^  =  _  4  (4«'  +  Jtt»). 

From  the  first  of  these  equations  we  find 


PLACE   IN   THE  ORBIT. 

The  second  equation  gives 

3(.J«5+i«')_wa» 

or,  substituting  for  a  its  value  just  found,  and  reducing. 


73 


a\3 


V^o  have  also 


(1  +  it«) 

^  1  +  «^ 


and  hence,  substituting  the  values  of  a  and  ^3  already  found,  and 
reducing,  we  obtain  finally 

A.(^1,^     \      1  2  0  8, ,9     I      10174  ,,n     I      IDfi,,"     1221  3i,15     I 82_„17^ 


(1  +  It')' 


Again,  we  have 


—  1, 


tan     U=  tan     (u  -{■  ai-\-  ^i^  -\-  yi^). 
Developing  this,  and  neglecting  terms  of  the  order  i^,  we  get 
tan-'  U=  tan- ' «  +  ^-1-,  (at  +  fii'  +  yi?)  -  T^-rzi  (<^'i'  +  2a,3i») 

Now,  since  w  =  tan^y  and  ?7=  tan  |F,  we  shall  have 


V^v  +  ~~-{ai^!ii'^ri'). 


2u 


^,-(aV  +  2a,Ji'')-f 


2(h'-0    ,;: 


or 


F_,  +  ._2^i  ,  /..2<? 2_a=«     \ 


at' I 


+ 


/     2r 


4ai3it        ,    2(it'— .p    3\  ., 


(72) 


(1  +  ti^y  '  (1  +  iC'f 

Substituting  in  this  equation  the  values  of  a,  /9,  and  y  already  found, 
aud  reducing,  we  o!)tain  finally 

^-"^        (i  J^,^f  *  +  (1  +  «»)!  ' 

8g,7     I     fl  2  8  8  ,.9    I     2  fi  H  8  1  ,,,11     I     4  fi  I  ,,I3    I     5  I  2  8  ,,15     I       »  0  4    ,,1T 

(.1  4-  ti'f 


\H^ 


74 


THEORETICAL   ASTRONOMY. 


IN!'   ■ 

,    P|i'  <!■ 

mr 


This  equation  can  be  used  wluinover  the  true  anomaly  in  the 
ellipse  or  hyperbola  is  given,  and  the  time  from  the  perihelion  is  to 
bo  determined.  Having  found  the  value  of  V,  we  enter  Table  \I. 
with  the  argument  T''and  take  out  the  corresponding  value  of  31; 
and  then  wc  derive  t  —  T  from 


'-'-Hil 


in  which  log  q,=:  9.96012771. 

For  the  converse  of  this,  in  which  the  time  from  the  perihelion  is 
given  and  the  true  anomaly  is  re([uircd,  it  is  necessary  to  express  the 
ditferenco  v  —  V  in  a  series  of  ascending  powers  of  i,  in  which  the 
coefficients  arc  functions  of  U.     Let  us,  therefore,  put 


u 


u+o:i  +  [i'i'  +  r'i^  +  &c. 


Substituting  this  value  of  u  in  equation  (70),  and  neglecting  terms 
multiplied  by  i*  and  higher  powers  of  /,  wc  get 


U-\-\U'  +  W{\+U^)-lU'-tU')i 


kit—  T)V\  +  e 

+  f/S'l  1  +  C7')  -^  Uo!^  —  2  f/V  (1  +  C7»)  +  3  [75  +  3  u^)  {t 
+  (/  (1  +  U')  +  I  a"  +  2  Ua'fi'  +  3  UV  (1  +  [/»)  —  2/?  C/Hl  +  U') 

—  4  U'a"  —  2  Ua"  —  4  [/'  —  I  f/")  i\ 

But,  since  the  first  member  of  this  equation  is  equal  to  U-\-  ^U^,  we 
shall  have,  by  the  i)rinciple  of  indeterminate  coefficie    3, 

a'(l+   U')-'^U'—%U'=^0, 

fi'  (1  +  U')  +  Vo!'  —  2  U'o!  (1  +  t7«)  +  3  [75  _|_  a  jj^  ^  q, 

/  (1  +  V)  +  lo!'  +  2  LV,5'  +  3  U*o:  il+V')—  2,}'  U\l  +  U') 

_  4  u'o!-'  —  2  Ua'-'  —  4  f/'  —  j  f/»  =  0. 

From  these  equations,  we  find 

fC7^+|£7» 

'  ~~  ("1  +  U'f 

292  rri   I   70  2  8  rr»  I    loa^e  rni   i    is2  7"r"_i_  fifi92  /"T'lsi     i  84_  Tin 


If  we  interchange  v  and  Fin  equation  (72),  it  becomes,  writing  a',| 
/9',  r'  for  a,  ^,  r, 


A  = 


St 


wherein  s  e: 

length  of  arc 

Wo  shall,  th 

Wlien  X  = 


PLACE   IN   THE  ORBIT. 

2tt'      .   .  /     2,5'  2a''U 


76 


-1^1       -       •   i/     2,5'  2a''^r     V 

,  /  _2/ ^4a',S'  r/      ,  2_(  f/^-  J)    ,  \ 

"^  \  1  +  f7»       "(1  +  U'?  "^  (1  +  U'f       I 


Substituting  in  tliis  cMjuation  the  above  values  of  a',  ^9',  and  y\  and 
reducing,  we  obtain,  finally, 

by  moans  of  M'hicli  v  may  be  determined,  tlic  angle  F  being  taken 
from  Table  VI.,  so  as  to  correspond  with  the  value  of  31  derived 
from 

Equations  (73)  and  (74)  arc  applicable,  without  any  modification, 
to  the  case  of  a  hyperbolic  orbit  which  differs  but  little  from  the 
parabola.  In  this  case,  however,  e  is  greater  than  unity,  and,  conse- 
quently, i  is  negative. 

28.  In  order  to  render  these  formulte  convenient  in  practice,  tables 
may  be  constructed  in  the  following  manner: — 
Let  x=-^v  or  V,  and  tan  \x  =  6,  and  let  us  put 


^=:_if+i^ 


l^y 


B 
B' 


luo(i  +  w'y 

2^^    I     5  0  8^7     1       8fi^fl9_l       18^Y?'> 

ioooo  (i  +  tf^j* 


s, 


10000(1  +  tfO* 

S84rt7    I      97r>-Jin»    I      37  S  i  8/Jll  _,      lfi48/in    I      17fi8/Jl5    I        181  fllT 
fi __  3  13"  T^  253;-.'^^   rl  I  7  r. ^_ i^  T57?>'^    ^7H7r,'^    T-fH7  5"     „ 

1000000(1  + ^■■')«  ' 

8fl7     I     5  2  8  ft/1»     I     2 «  3  8  1  flU     I      4  0  1  /ll3     i      5  12  8/JI5     I        9  0  4  fllT 

pi i"  T2B3d'^  i^  14  175^    i^  3  1.5^      r  7g7ft'^    n-  ^^^7^" 

1000000  (1  +  oy 

wherein  s  expresses   the   number  of  seconds  corresponding  to  the 
length  of  arc  equal  to  the  radius  of  a  circle,  or  logs  =  5.31442513. 
Wo  shall,  therefore,  have : — 
When  X  =  V, 


v=  F+  ^(lOOi)  +  5(1000'+  C(lOOi)'; 


76 


THEORETICAL  ASTRONOMY. 


niul,  when  x  =  v, 

V=v-A  (1000  +  £'(100/)'-  C'dOOO'. 

Tabic  IX.  gives  the  vahies  of  A,  B,  B',  C,  and  C  for  consecu- 
tive values  of  x  from  re  =^  0°  to  x  ■—  149°,  with  differences  for  inter- 
polation. 

When  the  value  of  v  has  been  found,  that  of  r  may  be  derived 
from  the  formula 

\  -\-  e  cos  V 

Similar  expressions  arranged  in  reference  to  the  ascending  powers 

of  (1  —  e)  or  of  I  I  zr-—- 1  —  1  I  may  be  derived,  but  they  do  not  con- 

verge  with  sufficient  rapidity ;  for,  although  I  I  ■  I  —  1  I  is  less 

than  /,  yet  the  coefficients  are,  in  each  case,  so  much  greater  t'laii 
those  of  the  corresponding  jiowers  of  /',  that  three  terms  wil'  lot 
afford  the  same  degree  of  accuracy  as  the  same  number  of  teru.-  in 
the  expressions  involving  i. 

29.  Equations  (73)  and  (74)  will  serve  to  determine  v  or  t  —  T  in 
neai'ly  all  cases  in  which,  with  the  oi'dinary  logarithmic  tables,  the 
general  methods  fail.  However,  when  the  orbit  differs  considerably 
from  a  parabola,  and  when  v  is  of  considerable  magnitude,  the  results 
obtained  by  means  of  these  equations  will  not  be  sufficiently  exact, 
and  we  must  employ  other  methods  of  approximation  in  the  case  that 
the  accurate  numerical  solution  of  the  general  formuloe  is  still  impos- 
sible. It  may  be  observed  that  when  E  or  F  exceeds  50°  or  60°,  the 
equations  (39)  and  (69)  will  furnish  accurate  results,  even  when  e 
differs  but  little  from  unity.  Still,  a  case  may  occur  in  which  the 
perihelion  distance  is  very  small  and  in  which  v  may  be  very  groat 
before  the  d'  ippearance  of  the  comet,  such  that  neither  the  general 
method,  nor  the  special  method  already  given,  will  enable  us  to  de- 
termine V  or  t  —  T  with  accuracy ;  and  we  shall,  therefore,  investigate 
another  method,  which  will,  in  all  cases,  be  sufficiently  exact  when 
the  general  formulae  are  inapplicable  directly.  For  this  purpose,  let 
us  resume  the  equation 


h{t—T) 


=  E — e  sin£', 


which,  bint 
kit-  2 

If  we  put 

we  shall  ha 

k{t- 

Let  us  no 

and 

then  we  hav( 


When  B  is  k 
be  derived  di 


a" 


and  then  fron 
to  find  the  va 
of.l. 
Now,  we  hi 


and  if  we  put 

sin 

We  have,  al 

E= 


nsccu- 
intor- 

erivod 


powers 
ot  con- 


31'  than 
V\\}  not 
erni>  in 


-Tin 
cs,  the 
erably 
rcji  lilts 
exact, 

lase  that 
impos- 

jO°,the 
wheu  e 
ich  the 
y  groat 
general 
to  de- 
estigate 
t  wheu 
DOse,  let 


s 


PLACE  IK  THE  ORBIT.  77 

which,  >-incc  q  =  «(!  —  c),  may  be  written 

-^ ^ =  j^  {dE  4-  sm  £)  +  j^ .  -j-^  (^  —  siu  E). 


If  wc  put 
we  shall  luivo 


15 


j:—»mE 
^E+ainE' 

i  ,   1      1+de     A 


■  \)E 4-  sin E~       "*"  3  ■  6(1  —  e) 


Lot  us  now  put 


B  = 


» i«.  — 


tan'  ^t<; 


9Jg  4-  sin  .E 
20l/I 

l+9e 


then  we  have 


5(1 -e) 


A; 


r~T~ "  p- =  tan  Aw)  +  .J  tan' |i«. 

1/25'-^  -** 


(75) 


When  B  is  known,  the  value  of  w  may,  according  to  this  equation, 
be  derived  directly  from  Table  VI.  with  the  argument 

75^«-T)    v/T'a(l+9iy 

I  and  then  from  w  we  may  find  the  value  of  A.    It  remains,  therefore, 

to  find  the  value  of  B;  and  then  that  of  v  from  the  resulting  value 

of  .1. 

Now,  we  have 

2  tan  IE 


svaE-- 

I  and  if  we  put  tan^  \E  ==  r,  we  get 

2Ti 


1  +  tau'AJS' 


sin  E  = 
We  have,  also, 


1+^: 


2tJ(1— T  +  T»  — T»  +  &C.). 


E=1  tan  '  T^=  2t^  (1  —  ^r  +  ^t»  -  ^t^  +  &c.). 


78 


TIIEOKETICAI.   ASTRONOMY. 


Thoroforc, 

15(JS;— 8m£)  =  2r^(10T-7T'-{-  7r'-igo-i  +  &c.), 
ami 

9E  +  sin  E  =  2t^  (10  —  'j^r  +  '/t'  —  V^'  +  V^  —  &c.). 
Hcneo,  by  division, 

IP sin  7? 

^^9£+"8iui;  """*""  o^+35^  20Sa^    +74.1376- 

1  OHOnr,  8  8r«  4-  Arp    • 

2T80ffS76~  T^  **'^'» 

and,  inverting  this  series,  we  get 

A 

^   —  ■••         5-^  T^  T75-^   n-  525-^    i^  138S7S^*    T^  TiiaSliiS-^   ^  *^'^-> 

which  converges  rapidly,  and  from  which  the  value  of  —  may  l)e 
found. 

Let  us  now  put 

then  the  values  of  C  may  be  tabulated  with  the  argument  A;  and, 
besides,  it  is  evident  that  as  long  as  A  is  small  C'^  will  not  differ 
much  from  1  +  ^A. 

Next,  to  find  JS,  we  have 

^  —  ^    Ki-  —  S^  -r  175^  —  525^   i^  TDTDsljB^  *^^-.)> 

and  hence 

T7-J. —  ■"  —  -^  T  T7-5^         35-J-6^  +  3  3<iS7r.^         <*C., 

from  which  we  easily  find 

If  we  compare  equations  (44)  and  (56),  we  get 
tau  ^^=  l/^^tan  ^F. 

Hence,  in  the  case  of  a  hyperbolic  oi'bit,  if  we  put  tan^^i^=  t',  we 
must  write  —  r'  in  place  of  r  in  the  formula;  already  derived ;  and, 
from  the  series  which  gives  A  in  terras  of  r,  it  appears  that  A  is  in 
this  case  negative.    Therefore,  if  we  distinguish  the  equations  fori 


liypcrlioli 

and  ("  ii 

1        A' 

I"'       r'  ' 

Table  : 

and  the  h 

For  every 

and  (69)  n 

TJie  equ 

gives 

or,  substitui 

The  last  ( 
Hence  we  d( 


Tlio  cquatioi 
a*^  (76),  the  a 
from 


I  For  tlie  radii 
[lastof  equati 


When  t-~ 
P=l,  and  e; 


Pr.ACE   IN   THE  OllBIT. 


79 


Iiv|)orl)olie  motion  from  thosf  for  elliptic  motion  by  writing  A',  B', 
and  ("  in  place  of  A,  B,  and  t',  rc'.s[)(;ctively,  wc  shall  have 

1        A' 

.O   — -^  T^  T7S-^     —  dJS-^     i^  3;10  8  71V-^*     — **^' 

Table  X.  contjiins  the  values  of  log  B  and  log  C  for  the  ellipse 
and  the  hyperbola,  with  the  argument  A,  from  ^l  =-=  0  to  ^^l  -  0..'}. 
For  every  ca.se  in  which  A  exceeds  0.3,  the  general  formulte  (.'59) 
and  (G9)  may  be  conveniently  applied,  as  already  stated. 

The  equation 


tan 


i»=V| 


+  e 


tan.'.E 


gives 


tan'>  =  ^-t-^  AC, 
-*        1  — e         ' 


ad;  and, 

It  it  is  in  I 

tions  for 


or,  substituting  the  value  of  A  in  terms  of  w, 


tan  iv  =  Ctan  hv  \  ~z — r-—-. 
-     \   1  -}-  9e 

The  last  of  equations  (43)  gives 


Hence  we  derive 


rco^lv  =  qco,H.E=^-^. 


{l+AC'^co&'-^v 


(76) 


(77) 


The  equation  for  v  in  a  hyperbolic  orbit  is  of  precisely  the  same  form 
as  (7G),  the  accents  being  omitted,  and  the  value  of  A  being  computed 

from 

4  =  ^/^tan'>  (78) 

For  the  radius-vector  in  a  hyperbolic  orbit,  we  find,  by  means  of  the 
I  last  of  equations  (63), 

(79) 


(1— ^C")cos4v 


When  t  —  T  is  given  and  r  and  v  are  required,  we  first  assume 
r',  we  H.B=  1,  and  enter  Table  VI.  with  the  argument 


M= 


_q,((-r)T/,'o(i+96) 

^i       '  B         ' 


80 


TirEORETICAI.   ASTROXOMY. 


in  whieli  log  ('^,  -  =  9.9()()12T71,  and  take  out  the  corrospoiuling  value 
of  'w.     Tliea  wo  derive  A  I'rum  the  equation 


A 


5ri 


^'K         3  1 

tan  -Aw, 

1  -r  9e 


in  the  ca^e  of  the  ellipse^  and  tVonx  (78)  in  the  case  of  a  hyperbolic 
orbit.  \V'ith  the  resulting  vahie  of  A,  we  find  from  Table  X.  the 
oorre,-J]ionding  value  of  log  J5.  and  then,  using  this  in  the  expi'ession 
ft)r  M,  Ave  repeat  the  operation.  The  second  result  for  ^1  ^vill  not 
require  any  further  correction,  since  the  error  of  the  first  assumption 
of  B  ■---  1  is  very  small ;  and,  with  this  as  argument,  we  derive  the 
value  of  log  (.'  fronx  the  table,  and  then  v  and  r  by  means  of  the 
equations  (76)  and  (77)  or  (79). 

When  the  true  anomaly  is  given,  and  the  time  t  —  T  is  required, 
we  first  compute  r  from 

tan^  h), 


T  -— 


H-e 


in  the  case  of  the  ellipse,  or  from 


tan''  hv, 


in  the  c-asc  of  the  hyperbola.  Then,  with  the  value  of  r  au  argu- 
ment, we  enter  the  second  part  of  Table  X.  and  take  out  an  a^jproxi- 
mate  value  of  ^1,  and,  with  this  as  argument,  we  find  logjB  and  log  (  . 
The  equation 

T 


A 


will  show  whether  the  ap})roximate  value  of  A  used  in  finding 
log  Cis  sufKciently  exact,  and,  hence,  Avhether  the  latter  requires  uny 
correction.     Next,  to  find  ?/,',  we  have 

^      ,         tanjc      I   i    I   9(5 
tan^t.^-^.^.^^-^; 

and,  with  ^o  as  argument,  we  derive  M  from  Table  VI.  Finally,  Wf 
have 

MBq^ 


i  •-  T- 


f;i/i^j(l  +  9^) 


(80) 


by  means  of  svhich  the  time  from  tlie  ]}erihelion  may  be  accurately 
determined. 


POSITION   TX    SPACK. 


81 


,-alue 


rbolio 
t.  the 

esslon 
11  not 
i\ptioii 
ve  the 
of  the 

:^uircd, 


J  argu- 
|)proxi- 
1  loir  C. 


iMuVing 
Ires  uny 


illy,  Avt 


(80) 


kiratcly 


30.  We  have  thus  far  treated  of  the  motion  of  the  heavenly  bodies, 
relative  to  the  sun,  without  considering  the  positions  of  tlieir  orbits 
insj)are;  and  the  elements  whieh  we  have  employed  are  the  eccen- 
tvicity  JU'-d  semi-transverse  axis  of  the  orbit,  and  the  mean  anomaly 
at  a  given  epoeh,  or,  what  is  e(|uivalent,  the  time  of  passing  tho 
|M'rilielion.  These  are  the  elements  whieh  determine  the  positif)n  of 
the  body  in  its  orbit  at  any  given  time.  It  remains  now  to  tix  its 
position  in  space  in  reference  to  .some  other  point  in  space  from  which 
we  conceive  it  to  be  seen.  To  accomplish  this,  the  position  of  its 
orliit  in  reference  to  a  known  plane  must  be  given;  and  the  Icmcnts 
which  determine  this  jmsition  are  the  longitude  of  the  periln  iion,  the 
longitude  of  the  ascending  node,  and  the  inclination  of  the  ])lanc  of 
the  orbit  to  the  known  plane,  for  which  the  plane  of  the  ecliptic  is 
usually  taken.  These  three  elements  will  enable  us  to  determine  the 
co-ordinates  of  the  body  in  space,  when  its  position  in  its  orbit  has 
been  found  I)y  means  of  tlie  tbrmuhc  already  investigated. 

The  hiiaitudr  of  the  a,^c('iid!nf/  node,  or  longitude  of  the  })oint 
tlirougli  which  the  body  passes  from  the  south  to  the  north  side  of 
the  ecliptic,  which  we  will  denote  by  Q,,  is  the  angular  distance  of 
this  point  from  the  vernal  equinox.  The  line  of  intersection  of  the 
plane  of  the  orbit  with  the  fundamental  plane  is  called  the  /hie  nf 

The  angle  which  tlic  plane  of  the  orbit  makes  with  the  i)lane  of 
the  eclii)tic,  Avhich  we  will  denote  by  /,  is  called  tla;  indlnatlon  of 
the  orbit.  It  will  readily  be  seen  that,  if  we  suppose  the  ])lane  of 
the  orbit  to  revolve  about  the  lino  of  nodes,  when  the  angle  /  exceeds 
1S<I°,  ^l  will  no  longer  be  the  longitude  of  the  ascending  node,  but 
will  become  the  longitude  of  the  descending  node,  or  of  the  i)oint 
through  which  the  planet  passes  from  the  north  to  the  south  side  of 
the  ecliptic,  wliieli  is  denoted  by  ?5,  Jind  which  is  measured,  as  in  the 
I'aso  of  fi4,  from  the  vernal  etjuinox. 

It  will  easily  l)e  understood  that,  when  seen  from  the  sun,  so  long 
as  the  inciinatiou  of  the  orbit  is  less  than  l)t)°,  the  motion  of  the 
liody  will  lie  in  the  same  direction  as  that  of  the  earth,  and  it  is  then 
-aiii  to  l»e  (Jirccf.  When  the  inclination  is  1)0%  the  motion  will  be  at 
liiiht  angles  to  that  of  the  earth;  and  when  /  exceeds  0()°,  the  motion 
in  longitude  will  be  in  a  direction  opi)osito  to  that  of  the  earth,  niid 
I  it  is  then  called  irtnu/rdde.  It  is  customary,  then^forc,  to  extend  the 
iiirlination  of  the  orbit  only  to  !)0^,  and  if  this  angle  exceeds  a  right 
jar.irh  ,  to  regard  its  supplement  as  the  inciinatiou  of  the  orbit,  noting 
-imply  the  distinction  that  the  motion  is  rdroyrade. 


82 


riir.Olil/IK  AI.    ASTUON'ONfy 


Till'  /(>ii>/lhi(lc  of  /he  /irrllirlioii,  which  is  denoted  hy  ~,  fixes  the 
]>()sition  of  the  orhit  in  its  own  plane,  and  is,  in  the  ease;  of  direct, 
Miolioii,  the  siMii  of  the  hiiifiitiuh'  of  tlie  ascendiii};'  node  and  the 
an^i'iilar  distance,  iiieasni'ed  in  the  direction  of  th(!  motion,  of  tlie 
perihelion  from  this  node.  It  is,  thc'reH>re,  tin!  an;j;nlar  distance  of 
the  perihelion  from  a  point  in  the  ori)it  whose  ant;iilar  distance  hack 
from  the  ascend i  11!^  node  is  c(pial  to  tlu;  lonj;itiide  of  this  node;  or 
it  may  lie  measured  on  the  ecliptic  (Vom  the  vernal  e(juino.\  to  the 
asceiidiiiu,'  node,  then  on  the  pliiiie  of  the  orhit  from  the  node  to  the 
place  of  the  perihelion. 

ill  the  case  of  retro|z;rade  motion,  tlw  lonifitndes  of  the  successive 
points  in  the  orbit,  in  the  direction  of  the  motion,  (h'crease,  and  the 
point  in  the  orhit  from  which  these  lon^'itiid(>s  in  the  orhil  arc  K  I 
mejisiired  is  taken  at  an  anti;ular  distance  from  the  aseendiii};  node 
c((iial  to  the  loni^itiidc  of  that  node,  hut  taken,  from  the  node,  in  tin.' 
same  direction  as  the  motion.  Hence,  in  this  <'ase,  tll(^  loiiLrittide  of 
tilt?  pci'ihclion  is  e(|iial  to  the  lont;itnde  of  the  ascentliiiii;  node  dimi- 
nished Itv  the  aiii!;iilar  distance  of  the  |)erilieIion  from  this  node. 


It 


may,  perhaps 


seem   desirai)li'   that    the  distinction-,  ilincf  and 


r(lr()(/ni<lr  motion,  should  he  ahaiidoned,  and  that  the  inclination  of 
tin'  orhit  should  he  measured  from  (f'  to  1<S0'\  sinc'c  in  this  <'as(> 
one  set  of  formula'  would  he  sulhcient,  while  in  the  common  form 
two  sets  are  in   part   retpiired.      IIow<'ver,  the  custom  of  astronoiiu'rs 


d   loi)i!,itude  of  the   perihelion   is  called  tli(^  iiiraii  /oiif/ifiidc,  an  ex- 


an 


pression  w 


hich 


can  occur  onlv  iii  tlie  case  o 


th 


)f  ellipt 


le  oroits. 


In  the  eas(!  of  retro<>rade  motion  the  lon<i'itude  in  the  orhit  is 
to  the  louj2;itU(h'  of  the  [X'rihelion  minus  the  true  anomaly. 


cqun 


.)!.  We  will  now  proceed  to  derive  the  forinuhe  for  dotcrminiiij]; 
the  co-ordinates  of  a  heavenly  body  in  .space,  wlien  its  position  in  its 
orhit  is  known. 

For  the  co-ordinates  of  the  j>osition  of  the  body  at  the  time  f.  wu 
have 

jf  — ;  r  sin  V, 


(lie  line  ( 
l:ikeii  at 
If  we 


(I)  \)v\l]<r    i 

jurilielioii 
ii'Diii  lh(!  1 
.N'<nv,  w 
.«       -  in 
M'coiiie 


(lie  Upper 
die  motion 

'lie  ''rt/iliilc 
i'fi      Its     I 

I'Imiics,  tlie 
t.'il^iii  as  tl 
Tlieii  we  sh 


seems  to   have  sanctioned   these  distiiictioiis,  and   (hey  may  be  per- 
petuated or  not,  as  may  seem  tidvantaiijeous. 

h'lirthcr,  we  may  remark  that  in  the  ciiso  of  direct  motion  the  suin 
of  the  true  anomaly  and  lonjjjitude  of  the  j)erilielioii  is  called  the 
Iriic  /<iii(/itii<lc  in   l/ic  orhit;  and  that   the  sum  of  the  im'tm  anomaly  H  || 


re  we  denot 
!'■  time  /,  b 


''onset  J II 


'•'"tti  lliese  \ 


"llich 


•serve  t 


POSITION'    IN    Sl'ACK. 


83 


;ui 


oqiuu 


rmmiii!' 


the  line  of  ii|)si<l('s  Ix'tiij^  taken  jis  {\w.  axis  of  .r,  and  iIk;  (tri^nii  Ix'inj^ 
t;il<cn  a(  (lie  ccnlrc  (if  the  sun. 

If  we  lake  I  lie  liii(!  of  nodes  as  the  avis  ol'  .r,  we  slwdl  have 

X  ■-    r  eo.s  ( I'  -t  '"), 

0)  \n-\u<r  the.  are  of  the  orhit  inforeejited  hetweeii  th((  piaee  of  tlie 
|)(rihelion  and  of  the  nixh',  or  tlie  an^idar  distance  of  the  jteriiie'ion 
from  lh(!  no<h'. 

N(iw,  we  hav(!  (o  ~  ^l  in  the  case  of  direet  motion,  and  a)  ~ 
!^l  n  in  tlie  cas(!  of  i'otvo<;rade  motion;  an<l  hence  the  hist  e<itiations 
hccome 

X  — .  r  cos  ( I'  :i:  -  rp  J^ ) 

2/  ~  r  sin  (vdtir.  ^l  ^l) 

tiie  upper  and   lower  sitrns  lieitii;  taken,   respectively,  accordinji;  as 

the  motion  is  direct  or  retrograde.     The  arc^  <?    i    -   ,    H        a  is  called 

tlie  'irf/uiiiri''    ■     fhr  Idt'itiuh'. 

Let    lis    iiiAv       'ler   the    position  of  the   hody   to  three   co-ordinate 

jiImiics,  the  origin   heiiig  at    the   centres  of  th(!  siin,  the  ecliptic  heing 

tiikcn  as  the   plane  of  .*'//,  and    the  axis  of  .c,  in   th(;   line   of  nodes. 

'riicu  we  shall  have 

x'       r  coH  v, 

y'  -     ±  /•  «iii  )/  cos  (", 

t'  =r.  ,•  ^iii  II  sin  /. 

If  we  denote  the  heliocentric  latitude  and   lnn<fitude  of  the;  hody,  at 
the  time  t,  by  b  and  /,  rcsjicclively,  we  shall  have 

af  =:^rmnf/  co.s (I  —  ft  \ 
if  r=r  cos  bm'ind  —  Q  ), 
z'  —  /-Hill  A, 


and,  0(>nse(jiH^n!tIy, 


1)11 


in  it-' 


C08«  -    COB  A/«>S  (I         Q,    , 

sin  V  cox  i  —  eos  h  *in(l       Q,). 
sin  H  nn  i      sin  /j. 


(H]  I 


hie 


^  we  H  From  tlnse  we  «lerivc 


tnn  I  /  —  ft  )  =  ±  tan  n  cos  /, 

ta*  'v  =  =t  tan  /'  niii  (I    -  ^  ), 


(m 


^y\w\\  serve  to  deterioiiM!  /  luid  h,  wli'  u  ^,  i/f  aun4  i  are  given,     hinco 


84 


THEORETICAIi   ASTROXOMY. 


coah  is  always  positive,  it  follows  that  I —  SI  smd  u  must  lie  in  the 
same  (luadrant  Avhcn  /  is  less  than  90°;  but  if  /  is  greater  than  90°, 
or  the  motion  is  i-etroj^rade,  /  —  ^  and  3(i0°  —  u  will  l)elon<>;  to  the 
same  (piadrant.  Ilenee  the  aml)i<>;uity  whieh  the  determination  of 
/—  ^  l)y  means  of  its  tangent  involves,  is  wholly  avoided. 

If  we  use  the  distinetion  of  retrograde  motion,  and  eonsider  i 
always  less  than  90°,  /  —  SI  and  —  u  will  lie  in  the  same  <|uadrant. 

32.  By  multiplying  the  first  of  the  equations  (81)  by  sin?*,  and 
the  second  by  cos  u,  and  combining  the  results,  considering  only  the 
upper  sign,  we  derive 


or 


cos  I)  sin  (u  —  /  +  $^ )  =:  2  sin  ?t  cos  «  sin^  U, 

cos  b  sin  (n  —  ^  -j-  SI)  '-^  sin  2u  sin'' ^'. 
In  a  similar  manner,  we  find 

cos  b  cos  (u  —  l-{-  SI)  =^  cos' it  -\-  sin' 11  cos i, 

which  may  be  written 

cos  b  cos  (u  —  /  -f-  $2  )  =  ,J d  -j-  cos  2u)  +10-  —  cos  2it)  cos  i, 
or 

cos  6  cos  (n  —  I  -{-  52  )  :=  A  (1  +  cos  i)  +  2  (1  —  cos  i)  cos  2u; 

and  henee 

cos  b  cos  (u  —  I  -j-  SI)  =  cos'  ;it  +  sin' U  cos  2u. 

If  we  divide  this  equation  by  the  value  of  cos  6  sin  («  —  I -{-  SI) 
alreadv  found,  wo  shall  have 


,         ,  ,    ^N  tan',U'sin2» 

tan («  —  /  +  Sl)~  ?— rr  %  ,  • ir- 

1  +  tan' Jit  cos  2)4 


fs.r, 


The  angle  u  —  /+  SI  is  called  the  rahii-tion  to  the  (w/lpfic;  and  tli 
cxj)ression  for  it  may  be  arranged  in  a  series  whiili  converges  rapi«ii\ 
when  /  is  small,  as  in  the  case  of  the  ]ilanet-.     In  order  to  effeei  this 
development,  let  us  first  take  the  e<puition 


tan  ij 


ti  sm  J- 

1  -j-  «  c«3a; 


Difi'ereiitiating  tiiis,  regarding  ?/  and  n  as  variables,  awl  i«<! 

find 

(hf  sin  a; 

(/;*       1  +  2"  cos  .i;  -f  'i 


;.  »M 


If     But  we  ha 
"lid,  putti 


TT*' 


rosiTiox  IX  SPACE.  85 

which  gives,  by  ilivlslon,  or  by  tlie  inetliotl  of  indotcrininatc  coofticients, 


dii 


=:  sin  X  —  /(  sin  2.v  -\-  n'  sin  .'Ic  —  )i^  sin  4.i-  -f  &c. 


(84) 


Integrating  tliis  expression,  wc  get,  since  y  ^=0  when  x  =  0, 

y  =  n  sin  x  —  In'  sin  2x  -j-  I )i^  sin  ox-  —  .{ n*  sin  4.c  + > 

which  is  tlie  general  form  of  the  deveh32)nu'nt  of  tlie  above  expression 
for  tan//.  Tlie  assumed  expression  for  tan  </ corresponds  exactly  witli 
tlie  formula  for  the  reduction  to  the  ecli])tlc  by  making  h  -  tan"  ii 
and  X  =  2u;  and  hence  Ave  obtain 

u  —  I  -\-  SI  -—  tan'^ },!  sin  2u  —  A  tan*  U  sin  4);  -|-  .'j  tan"  U  sin  <)» 

—  I  tan"  v]i  sin  8»  +  l  tan'"  ;U'  sin  lOu  —  &c.  (85) 

When  the  value  of  i  does  not  exceed  10°  or  12°,  the  iirst  two  terms 
of  this  development  will  be  sufficient.  To  express  u  —  /+  Si  i" 
seconds  of  arc,  tin;  value  derived  from  the  second  member  ot'  this 
equation  must  be  multiplied  by  20G2(j4.81,  the  number  of  seconds 
corresponding  to  the  radius  of  a  circle. 
If  Ave  denote  by  ii^  the  reduction  to  the  ecliptic,  avc  shall  have 

P     But  Ave  have  u  =  JI/+  the  equation  of  the  centre ;  lience 

I  =  M-}-  ~  -\-  equation  of  the  centre  —  reduction  to  the  ecliptic, 

inu],  putting  L  =  Il-r  z  ^^  mean  longitude,  we  get 

1:=  L  -{•  equation  of  centre  —  reduction  to  ecliptic.  (86) 

In  the  tables  of  the  motion  of  the  planets,  the  equation  of  the 
(viitre  (53)  is  given  in  a  table  Avith  M  as  the  argument ;  and  tlie 
icduction  to  the  ecliptic  is  given  in  a  table  in  Avhich  /  and  a  are  the 

iiruuincnts. 

•')'").  In  determining  the  place  of  a  heavenly  body  directly  from 
the  elements  of  its  orbit,  there  Avill  be  no  necessity  for  comi)uting  tlie 
I'duction  to  the  ecli])tic,  since  the  heliocentric  longitude  and  latitude 
'niiy  he  readily  found  by  the  formula'  {H'2).  When  the  heliocentri(! 
I  luce  has  l)een  found,  avc  can  easily  deduce  the  corrcspomling  geo- 
vntrie  place. 

Let  .r,  y,  z  be  the  rectangular  co-ordinates  of  the  planet  or  comet 
ii'tlrrfd  to  the  centre  of  the  sun,  the  plane  of  xy  being  in  the  ecliptic, 


86 


TJIEORETICAL   ASTRONOfV. 


tlu!  positive  axis  of  x  lioiiig  diroctod  to  the  venial  equinox,  and  the 
positive  axis  of  z  to  the  uortii  pole  of  the  eeliptie.     Then  we  shall 

have 

a;  :=  ?•  cos  h  cos  /, 
y  =^  r  cos  b  sin  /, 
z  =  )•  sin  h. 

Aj;'ain,  let  A'^  V,  Z  ha  the  co-ordinates  of  the  centre  of  the  sun  re- 
ferred to  the  centre  of  the  earth,  the  i)lane  of  A')"  being  in  the  eclip- 
tic;, and  the  axis  of  A'  being  directed  to  the  vernal  e([uinox;  and  let 
O  denote  the  geocentric  longitude  of  the  sun,  li  its  distance  from 
the  earth,  and  1'  its  latitude.     Then  we  shall  have 


x= 

-  E  cos  i'  cos  O , 

F  = 

li  cos  -  sin  O , 

Z  ^ 

:  li  sin  2'. 

Let  x',  y',  z'  be  the  co-ordinates  of  the  body  referred  to  the  centre  of 
the  earth;  and  let  /  and  ,5  denote,  res])ectively,  the  geocentric  longi- 
tude and  latitude,  and  J,  the  distance  of  the  planet  or  comet  from  the 
earth.     Then  we  obtain 


(87) 


a/  = 

=  J 

cos 

,?  cos  ?., 

?/  = 

=  J 

COS 

{i  sin  /, 

z'  ^ 

=  J 

sin 

/5. 

But,  evidently,  we  also  have 

X  —  .r  i-  A,  y 

and,  consequently. 


■y  +  Y, 


z  +  Z, 


4  cos  /?  cos  A  =  ?•  cos  />  cos  I  ~\-  li  cos  2"  cos  O . 
4  cos  /?  sin  ).  :-=  }•  cos  6  sin  /  +  ^^^  t'os  2"  sin  © , 
^  sin  li  =  r  sin  b  -\-  H  sin  2". 


(88) 


If  we  multiply  the  first  of  these  equations  by  cos  Q,  and  the  second 
by  sin©,  and  add  the  products;  then  multiply  the  first  by  sin  ©, 
and  the  second  by  cos©,  and  subtract  the  first  }>roduct  from  tin 
se(!ond,  we  get 


J  cos ,?  cos  {>.  —  ©  )  =  r  cos  h  cos  (I  —  © )  ]-  R  cos  2", 
J  cos  ,3'  sin  (A  —  ©  )  -;  r  cos  i  sin  ( /    -  © ), 
J  sin  ,3  =  r  sin  b  -\-  li  sin  2". 


(89 1 


It  will  be  (»]»served  that  this  transformation  is  equivalent  to  the  sup- 
position that  the  axis  of  x,  in  each  of  the  co-ordinate  systems,  is 


Ji'  we  sii 
tilde  is  Si,  . 
tioiis  (88)  b 


I'v  means  ot 

If  it  be 
dt'clination, 
values  of  fi 
ticn,  denotii 


now  ( 


POSITION    IN"   .SI'AC.'K. 


87 


directed  to  :i  point  whose  lon(>itii(le  is  ©,  or  tliat  the  system  liris  Ixun 
revolved  idiout  the  axis  of  -  to  a  lu'w  position  for  which  the  axis  of 
iihscissas  makes  the  an^^'h-  Q  with  that  of  tlie  jjrimitive  system.  We 
may,  therefore,  in  general,  in  order  to  effect  snch  a  transformation  in 
systems  of  e(piation,s  thns  derived,  simply  diminish  the  longitnde.-  l)y 
the  given  angle. 

The  ('(piations  (89)  will  (h'termine  /,  ,'i,  and  J  when  r,  h,  and  I  have 
heen  derived  from  the  elements  of  the  orbit,  the  qnantities  A',  0,and 
2'  k'ing  furnished  by  the  solar  tables;  or,  when  J,  ,5',  and  /  are  given, 
those  equiitions  determine  /,  h,  and  r.  The  latitude  -  of  the  sun 
never  exceeds  ±:  0".9,  and,  therelbre,  it  may  in  most  eases  l)e  neg- 
locted,  so  that  cos  2'  =:  1  and  sin  2'  -    0,  and  the  last  equations  become 


J  cos  /?  cos  (A  —  O  )  =  r  cos  h  cos  (/—©)  +  li, 
J  cos  /J  sin  (A  —  O )  =  V  cos  b  sin  (/  —  Q ), 
J  sin  /3  =  /•  sin  b. 


(90) 


If  we  suppose  the  axis  of  x  to  be  directed  to  a  point  whose  longi- 
tude is  SI,  or  to  the  a.scending  node  of  the  planet  or  comet,  the  e([ua- 
tions  (88)  become 


J  COS  I'i  cos  (A  - 

-SI) 

J  COS  /?  sin  (A  - 

-SI) 

J  sin  /J 

r  COS  u  +  A*  cos  -  cos  (©  —  SI), 
±  r  sin  a  cos  i  +  li  cos  1'  sin  (  ©  —SI),  C^l) 
r  sin  (/  sin  /  -f-  A*  sin  -', 


by  means  of  which  ,9  and  A  may  be  found  directly  from  SI ,  i,  >',  »iid  i'. 
If  it  be  required  to  determine  the  geocentric  right  ascension  an<l 
ik'clination,  denoted  respe(!tively  by  a  and  o,  we  may  convert  the 
values  of  /9  and  A  into  those  of  a  and  o.  To  effect  this  transforma- 
tion, denoting  by  e  the  oblic[uity  of  the  ecliptic,  we  have 

cos  '5  cos  a  =  cos  ,'1  cos  /, 

cos  (J  sin  a  =  cos  fi  sin  /  cos  s  —  sin  fi  sin  s, 

sin  S  =z  cos  ,3  sin  A  sin  s  -\-  sin  fi  cos  j. 


Let  us  now  take 


ami  we  shall  have 


n  sin  N  ■■=  sin  /J, 

n  cos  X  =  cos  (5  sin  A, 

cos  '5  cos  a  =t  COS  ;5  COS  X, 

cos  '5  sin  a  =-■  n  cos  (X  -(-  s), 
sin  '5  =  u  sin  ( N  -\-  s). 


88 


TIIEOKETICAIi   ASTRONOJIY. 


Therefore,  we  ohtain 


ttiniV; 


tail  /J 

8111  / 

tun  0 


C08(iV+0^        , 

tan  o  =  tan  /, 

eos  N 


(92) 


We  also  Iiave 


tan  (xY+  ')  f'i"  <*• 
cos  (N  +  e) 


COfi. 


cos  0  sin  o 
cos  (J  sin  A 


which  will  servo  to  cheek  the  calculation  of  a  and  d.  Since  cos  o  and 
cos;9  are  always  positive,  cos  a  and  <'os/  must  have  the  same  sign, 
and  thus  the  (juadrant  in  whicii  a  is  to  he  taken,  is  determined. 

For  tile  solution  of  the  inveise  problem,  in  which  a  and  i)  are 
given  and  the  values  of  /  and  ,5'  are  reipiired,  it  is  only  necessary  to 
interchange,  in  these  equations,  a  and  /,  o  and  f-i,  and  to  write  —  e  in 
})lace  of  £. 

34.  Instead  of  pursuing  the  tedious  process,  Avhen  sev(!ral  places 
are  ix'cpiired,  of  computing  first  the  heliocentric  place,  then  the  geo- 
centric; place  referred  to  the  ecliptic,  and,  finally,  the  geocentric  rigiit 
ascension  and  declination,  we  may  derive  formultc  which,  when  cer- 
t'.iiu  constant  auxiliaries  have  oiu^e  been  computed,  enable  us  to  derive 
tlu!  geocentric  place  directly,  referred  either  to  the  ecliptic  or  to  the 
e([uator. 

We  will  first  consider  the  case  in  which  the  ecliptic  is  taken  as  the 
fundamental  plane.     Let  us,  therefore,  resume  the  equations 

x'  =  r  cos  )/, 

y'  =:-  ±:  r  sip  :i  cos  i, 

z'  =  r  sin  u  sin  !, 

in  whicii  the  axis  of  ;k  is  supposed  to  be  directed  to  the  ascending  node 
of  the  orbit  of  the  body.  If  we  now  pass  to  a  new  system  c,  i/,  z, — 
the  origin  and  the  axis  of  z  remaining  the  same, — in  which  the  axis 
of  .(;  is  directed  to  the  vernal  equinox,  we  shall  move  it  back,  in  ;i 
negative  direction,  e([ual  to  the  angle  Q,,  and,  consequently, 

X  =  ;(;'  cos  ^  —  !/'  sin  ^, 
y  =  x'  sin  £1  +  y'  cos  Q, , 


Therefore,  we  obtain 


X  =  r(cos  ?f  cos  Q,  =F  sin  u  cos  t  sin  Q,), 
y  z:zzr{-V2  sin  u  cos  t  cos  Q,  +  ^-'os  if  sin  Q, ), 
g  =  r  sin  a  sin  i, 


(93) 


POSITION    IN    SI'ACIK. 


89 


which  arc  the  cxprt'ssions  for  the  heliocentric!  co-ordinates  of  a  planet 
(W  ((Pinet  referred  to  tlu;  ecli|)tic,  the  positive  axis  of  ./■  I)i'in<f  directed 
to  tlie  vernal    e(|uinox.     Tiie    upper  si<j,ii   is   to  he  u>ed   when    the 
motion  is  direct,  and  the  lower  sign  when  it  is  retrograde. 
Let  us  now  put 

cos  Q  -   sinrt  sin  A, 

^  cos  /  sin  £1  '-■  sin  n  cos  A, 

sin  Q,  n    sin  />  sin  l>, 

±  cos  /  cos  Q,  =^^  sin  h  cos  Ji, 


(04) 


ill  which  sin  a  and  sin  6  are  positive,  and  the  expressions  for  the  co- 
ordinates become 

a;  =  r  sin  a  sin  (A  -f-  n), 

y  =zr  sin  h  sin  (/>'  -\-  u),  (JM) 

z  =r  sin  i  sin  ». 

The  auxiliary  fpiantities  (t,  h,  A,  and    />,  it  will  be  observed,  are 
functions  of  Q,  and  /,  and,  in  computing  an  ephenieris,  are  constant 
?n  long  as  these  elements  ai'e  regarded  as  constant.     They  arc  called 
die  i-niiKUinfx  for  the  ecliptic. 
To  deternunc  them,  avc  have,  from  ecpuitions  (94), 


cot  A^=:  -\-  tan  Q,  cos  /, 
cos  Q, 
sin  ^1 ' 


sui  a 


cot  B  =  ±:  cot  Q  cos  i, 
."in  ^ 
ain  Ji' 


sin  b 


the  upper  sign  being  used  when  the  motion  is  direct,  and  the  lower 
sign  when  it  is  I'ctrograde. 

The  auxiliaries  sin  a  and  sin  b  are  always  positive,  and,  therefore, 
sin.l  and  cos  Q,  sin  /^  and  sin  SI,  resj)ectively,  must  have  the  same 
i^igiis.  The  quadrants  in  which  A  and  B  are  situated,  are  thus  deter- 
mined. 

From  the  equations  (94)  we  easily  find 


cos  a  =-  sin  t  sin  ^, 
cos  b  ==  —  sin  ('  cos  £1. 


(96) 


If  we  add  to  the  heliocentric  co-ordinates  of  the  body  the  co-ordi- 
nates of  the  sun  referred  to  the  eai'th,  for  which  the  ecpiations  have 
already  been  given,  we  shall  have 


X  -f-  A'  =^  J  cos  (J  cos  f., 
y  4-  y  ---  -1  cos  ,3  sin  A, 

z  -\-  Z  =  J  sin  fi, 


(97) 


90 


TII KO  It  KT  K  A  I ,    A  ST  IK  )NOM  Y. 


^vlli(•ll  siillicc  to  (Ictcnii'mL'  /,  ^-i,  mikI  J.     The  values  of  a  mid  o  mav 
be  derived  iVom  tlicsc  by  lucan.-^  dl"  tlic  ('»|iiati(iii,s  (!)2). 

.'};").  W'c  shall  now  (Icrivc  (lie  roriiiida'  for  (Ictcnniiiinji'  a  and  H 
diriTtly.  I'^or  (his  purpn.-c,  let  .c,;/,:.  Ix'  tli''  heliocentric  co-onliiiatcs 
of  the  body  referred  to  the  ecpiator,  the  jMisitive  axis  of  ,(•  beinjf 
directed  to  the  vernal  e(|iiino.\.  To  pass  from  the  system  of  co- 
ordinates referred  to  the  ecliptic  to  those  rel'erred  to  tlu;  ('(piator  as 
the  fundamental  |)lane,  we  must  revolve  the  systi'm  nejiatively  aroinid 
the  axis  of  ./•,  so  that  the  axes  of  :■  and  //  in  the  new  system  make 
the  angle  s  with  those;  of  the  primitive  system,  £  being  the  obli(piity 
of  the  eellptie.     In  tliisi  easi',  we  have 

x"  =  X, 

ij'  =  y  cos  e  —  2  sin  s, 

z"  =  y  sin  s  +  -  f  O''  -• 

•Substituting    for  x,  //,  and  -  their  value.s  from    equations  (93),  aud 
omitting  the  aeeenis,  we  get 


X  =  r  cos  n  cos  ^  =p  /•  sin  n  cos  /  sin  J^ , 

y=:^r  cos  II  sin  £1  con s  -\- r  s'm  n  (±  cos/  cos  Q,  coss  —  sin i  sine), 

g  =  r  cos  i(  sin  £1  sin  £  +  ''  ^'"  "  ( ^  ^'^^'^ '  t'0'">  Q>  >^'i^  -  +  i^'ii  i  cos  s). 


(08) 


These  arc  the  expressions  for  the  heliocentric  co-ordinates  of  tlie 
])lanct  or  comet  referred  to  the  ecpiator.  To  reduce  them  to  a  con- 
venient I'orm  I'or  numerical  calculation,  let  us  put 


cos  SI  =  ^in  «  sinvl, 

zp  COS  !■  sin  J^  ;-=  sin  a  cos^, 

sin  Q  cos  e  ==  sin  b  sin  B, 

dh  cos  i  cos  SI  cos  e  —  sin  /  sin  e  =:=  sin  b  cos  Ji, 

sin  Si  sin  s  =:=  sin  c  sin  C, 

rt  cos  i  cos  SI  f"'"  ^  +  ^i»  >■  cos  £  =-  sin  c  cos  C; 

and  the  expressions  for  the  co-ordinates  reduce  to 

a;  =  r  sin  a  sin  (A  -\-  v), 
y  =  r  sin  b  sin  {B  j-  n), 
z  =  r  sin  c  sin  (  C  -\-  n). 


(99) 


(100) 


The  auxiliary  quantities,  a,  b,  c.  A,  B,  and  C,  are  constant  so  long 
as  SI  tiiid  /  remain  unchanged,  and  are  called  consUmtn  for  the  cquoiof. 

It  will  be  observed  tliat  the  equations  involving  a  and  A,  regaid- 
ing  the  motion  as  direct,  correspond  to  the  relations  between  tin 
parts  of  a  quadrantal  triangle  of  which  the  sides  are  i  and  a,  tlu 


I'OSITIOX    ]N    SI'Al'K 


01 


aiijrlt  iiH'Iudcd  hotwccn  tlieso  sides  lu'injf  that  wliicli  wo  dcsiji'iiatc  l>y 
.1,  and  tlu!  aii;i;lc  opposite  tiic  side  o  hciiio-  !)()"-  J^.  In  the  case; 
lit'  li  and  />,  tiic  relations  are  those  of  liie  |)arts  of'a  spherical  triant;le 

s,  II  lu'inn'  the  an^le  iiichnled 


which  the  sides  are  /;,  /,  and  !*(»' 


Iiv  /'  and  //,  an( 


iSd 


9,   tl 


e  anji'le  (tpposite  tlio  su 


te  th 


de  h.     Finther 


HI 


liie  case  of  c  and    (',  the  relations  are  those  of  the   parts  of  a 


1  tl 


rie  of 


wliK'li  the  sides  are  c.  i,  aii» 


U,  tl 


le  anir 


\v  ('\ 


)Cllll 


fijilicneal  triang 

tliiit  included  hy  the  sides  /  and  c,  and  1(S()°        Q,  that    included   Ky 

the  sides   /  and   s.      We   have,    therefore,   the    followiii";    aiKlitioiial 


(■(|iiatiuns 


cosrt  --ir;  sin  /  sill  Q,, 

cos  6  =  —  cos  J2  sin  /  cos  e  —  cos  /  sin  e, 

cos  c  =  —  cos  Q,  sin  /  sin  j  -j^  cos  i  c 


(101) 


OS 


In  the  case  of  retrofirade    motion,  we   must   substitute    in    tlieso 


isd- 


/  in  place  of  / 


The  <;'eometrical  sifj;iiification  of  the  auxiliary  constants  for  the 
('(Hiator  is  thus  made  apparent.  The  an<i;Ies  a,  h,  and  c  are  those 
wiiiiji  ;i  line  drawn  I'roni  the  orij^in  of  co-ordinates  perpendicular  to 
the  plane  of  the  (jrhit  on  the  north  side,  makes  with  the  j)ositive  co- 

1  A,  11,  and  ('  are  the  anjjles  which 


on 


Imate  axes,  res])ec 


tivcb 


aiK 


the  three  planes,  passin«r  thi'oiii>h  this  line  and  the  co-ordinate  axes, 
iniikcwith  a  plane  passing  through  this  line  and  perpendicular  to  the 
line  of  nodes. 

Ill  order  to  facilitate  the  computation   of  the  eonstants   for  the 
equator,  let  us  introduce  another  auxiliary  quantity  E^,  such  that 

sin  i  =:  e^  sin  E^, 
±:  cos  i  cos  ^2=^0  cos  £„, 

e^  hoing  always  positive.     We  shall,  thereibre,  have 


tan  E„ 


tan  i 


cos 


9>' 


iSiiK'o  both  Ty  and  sin/  are  positive,  the  angle  T?,,  cannot  exceed  180°; 
and  the  algebraic  sign  of  tan  A',  will  show  whether  this  angle  is  to 
1)0  taken  in  the  first  or  second  (piadrant. 
The  first  two  of  equations  (99)  give 


ami  the  first  gives 


cot  ^4  =  -+- '  uu  Q,  cos  ii 


sm  a  = 


cos  Q, 
siu^' 


-.% 


^, 


%»^^V^( 


IMAGE  EVALUATrON 
TEST  TARGET  (MT-S) 


1.0 


I.I 


lii|^8     |2.5 

•^  ni   |2.2 

1^  i: 

III 


t  Ui    12.0 


L25  III  1.4 


I 


1.6 


y] 


/: 


^;^ 


>^ 


Photographic 

Sdences 

Corporation 


33  WIST  MAIN  STRiiT 

WHSTIR.N.Y.  I4SS0 

(716)  873-4503 


«^ 


92  TIIKOKKTlCVL   AHTIIONOMY. 

From  tlu!  fourth  of  (fiuations  {{)[i),  introiluciiig  c^  and  A'„  we  get 
sin  h  cos  B=^  e„  cos  E^  cos  c  —  r,  hIii  E^,  m\  t  =-  e^  cog  (ii'^  -f  c ). 
sill  b  siu  iy  —  iiiii  ft  cos  t ; 


But 

therefore 
coti? 


HIU  ft 

We  have,  also, 

gin  6 

In  a  siniihir  manner,  we  find 


COS  (E„  +  t) co8  i  cos  ( Ef^  -f- «) 

tan  ft  cos  A'u  cos  c 


cose 


sin  ft  ccic 
sin  ii 


cot  C'= 


and 


cos  t  sin  (Eg  4-  •) 

tan  ft  cox  Aj  '        sin  e       * 

sin  ft  sin  c 


8mc  = 


sin  C 


The  auxiliaries  sino,  sin  6,  and  sin  a  are  always  positive,  and,  then- 
fore,  sin -.4  and  cos  ft,  sin  B  and  sin  ft,  and  also  sin  ('  and  sin  JJ, 
must  have  the  same  sijjns,  which  will  determine  the  (|uadiiint  in 
which  each  of  the  anjjles  A,  Ii,  and  ('is  situated. 

If  we  multiply  the  last  of  ec|uations  (911)  hy  the  third,  and  tlu 
fifth  of  these  cfjuations  hy  the  fourth,  and  sui)ti'act  the  first  prodiut 
from  the  last,  we  get,  hy  rctlnction, 


But 


sin  b  sin  c  sin  {C —  JS)  =  —  sin  t  sin  ft. 
sin  a  cos  ^1  =  =P  cos  t  siit  ft ; 
and  hcnec  we  derive 

sin  6  sin  r  sin  ( C —  B) 


sin  a  cosvl 


=  ±  tan  I, 


which  serves  to  cheek  the  accuracy  of  the  nnmrrieal  computation  of 
the  constants,  since  the  value  of  tan  /  ')l)taine<l  from  this  fon.iulaj 
must  agree  exactly  with  that  usetl  in  the  calculation  of  the  values  ct 
these  constants. 

If  we  jmt  A'  =  A±ir-  Q,  B'  =  B±7:r^  ft,  and  C  =  C ± 
^^  ft,  the  upper  or  lower  sign  being  use<I  according  as  the  Motion  ijj 
direct  or  retrograde,  we  shall  have 


Pf>8ITIO.\    IN'   SI'AfE. 

«  =  r  pin  It  ,«in  '  A'  -f  r), 
y  =  rniii  //sill  (  If  -\-  r), 
s  =z  r  .xiiir  sir  (C  -\-  v), 


(102) 


a  transforinntion  wliirh  is  pcrlmps  unnccrMsnry,  hut  wliicli  i.s  oon- 
viiii'iit  wlii'ii  a  M-rii's  of  phircs  is  t<t  !)(>  cuinjiiitcd. 

It  will  Ik'  oltsorvwl  that  thi'  j'oniiuhi'  tnr  comimtiiin  tlic  constants 
II.  i,  «',  A,  li,  and  (\  in  the  (Uso  of  direct  motion,  arc  convcrtj-il  into 
tlio-c  for  the  case  in  which  the  distinction  of  retroj^nuU;  motion  i.s 
atlopted,  hy  simply  usinj;  180°  —  /  instead  of  /. 

36.  When  the  heli<K't'ntric  co-ordinates  t>f  the  hody  have  Imhmi 
foiiiul,  relcrred  to  the  e<|iiator  as  the  fniidamental  phme,  if  we  a(Ul  to 
tlu'M'  the  jjeocentric  co-ordinates  of  tiic  sun  referred  to  the  same 
fiiiiilaiiicntal  ph»n<',  the  sum  will  he  the  geocentric  co-ordinates  of 
the  ImmIv    eferred  also  to  the  e<|iiat<M'. 

For  the  co-ordinates  of  the  sun  rjfcrred  to  the  centre  of  the  earth, 
we  liuve,  neglecting  tli<>  latitude  of  the  sun, 

Y=  It  m\  O  eose, 

Z  =  H  sin  O  sin  s  —  i'tnn  r, 

ill  wliicii  R  represents  the  radius-vector  of  the  earth,  ©  the  sun's 
](iii;ritu<U',  and  £  the  ohli(|uity  of  the  ecliptic. 
We  shall,  therefore,  have 


X  -|-  A'  -=  J  cos  ii  cos  o, 
y  -\-  )'  =  J  cos ')  sin  a, 
z  ~\-  Z  =  J  sin '5, 


(t03) 


wliicji  suffice  to  determine  a,  d,  and  J. 

\\'  we  have  regard  to  the  latitude  of  the  sun  in  computing  its  geo- 
centric co-ordinates,  the  formuhe  will  evidently  hecome 


X=Ji  cos  O  cos  -, 

Y=  R  sin  O  <*•»«  -  <'os  £  —  li  sin  -  sin  c, 

Z=  M  sin  O  cos  ^'m\  t-  -f-  J{  sin  -cos  t, 


(104) 


in  wliieh,  since  2'  can  never  excee<l   ±  0".9,  cos  2'  is  very  nearly 
'■<|ii:il  to  1,  and  sin  -     -  -. 

'flic  longitudes  and  latitudes  of  the  sun  may  he  derived  from  a 
Milar  cpliemeris,  or  from  the  solar  tahies.  The  princi|>al  astrononiicjd 
cplicincrides,  such  as  the  lirrlhirr  Axtrnnomittchfn  Jahrhuch,  the 
Xdulii'dl  Alinamic,  and  ihc  .'.meyican  Epheuurls  and  Nauticxd  Al- 


Til Fy»RETICAL   ASiTRONOMV. 


mnnnc,  rontain,  for  cnch  year  for  whicli  they  arc  puMi'sluKl,  (be 
o(|iiat«>rial  co-onliiiatcs  of  tho  .smi,  rcfrrrcil  both  to  the  mean  o(|uin(tx 
and  ('(juator  of  the  l>o^iniiingof  thoyt-ar,  and  to  the  apparent  equinox 
i)i'  the  date,  taking  into  aceonnt  the  latitude  of  the  .sun. 

.'57.  Tn  the  cjise  of  an  oiliptic  orhit,  we  may  determine  the  co- 
ordinates (lireetly  from  the  eeecntrie  anomaly  in  the  following 
maimer : — 

Tho  equations  (102)  give,  aeeenting  the  letters  a,  b,  and  c, 

x  =  r  cos  V  sin  n'  sin  A'  -f  r  sin  v  sin  «'  cos  A', 
y  =  r  cos  V  sin  b'  sin  If  -\-  r  sin  v  sin  h'  cos  If, 
«  =  r  cos  r  sin  c'  sin  C"  -f  r  sin  v  sin  c'  cos  C". 

Now,  since  r  cos  y  —  o  coa  E  —-  ae,  and  r  sin  v  ~  a  cos  ^  sin  E,  we  shall 
have 

a;  =  a  sin  a'  sin  A'  cos  E  —  ac  sin  n'  sin  A'  ■■{-  a  cos  f>  sin  «'  cos^'  sin  E, 
y  =  «  sin  b'  sin  /^  cos  E  —  ac  sin  //  sin  Ji'  -\-  a  cos  y  sin  6'  cos  If  sin  A', 
8  =  a  sin  c'  sin  t"  cos  E  —  ae  sin  c'  sin  C"  -f  a  cos  y  sin  c'  cos  C  sin  £". 

Let  us  now  put 

a  cos  f  sin  «' cosyl' =^  ^,  cos //„ 
o  sin  a'  sin  A'  =  /,  sin  />„ 

—  «<*  sin  a'  sin  vl'  =  —  ei.^  sin  L,  =  w, ; 

a  cos  y  sin  6'  cos  If  =  kj  cos  />,, 
rt  sin  //  sin  If  :=  -1,.  sin  L,, 

—  ac  sin  6'  sin  /i*  =  —  ^ ^^  sin  L,  =  Vj-, 
a  cos  f  sin  c'  cos  C"  =  A,  cos  X„ 

o  sin  c'  sin  (  "  =  ^,  sin  />„ 

—  ac  sin  c'  sin  C"  =  —  e^,  sin  L,  =  v, ; 

in  which  sin  a',  sin  6',  and  sin  c'  have  the  same  values  as  in  equations 
(102),  the  accents  l)eing  ad<led  simply  to  mark  tlie  necessary  dis- 
tinction   in   the   notation   employed   in  these  formula;.     Wo  shall, 

therotbre,  have 

a;  =  A,  sin  (X,  -\-  E)  +  v„ 

y  =  ij  Bin  (Lj  -f-  E)  H-  v„  (lOo) 

8  =  ^,  sin  (L,  4"  -E)  +  *'f 

By  means  "f  these  formula:',  the  co-ordinates  are  found  directly 
from  the  eccentric  anomaly,  when  the  constants  ^„  Xy,  /(„  Z„  Zy,  L„ 
v„  v,.,  and  v„  have  been  computed  from  those  already  found,  or  from 
«,  h,  c,  -4,  if,  and  C.    This  method  is  very  convenient  when  a  great 


rasiTioN  IN  SPACE.  m 

miinl)or  of  pfcocontric  plaroH  aiv  to  he  eonipiitcd  ;  l)iit,  wlioii  only  ii 
I'l'W  pliUTH  arc  rt'tniinM.!,  the  achlitioual  hihor  of  coiiipiitin^  s(»  many 
auxiliary  (|imiitities  will  not  he  coinjM^nsate*!  hy  the  facility  alfonled 
in  tlic  numerical  calculation,  when  these  constants  have  hcen  deter- 
mined. Further,  when  the  ephenieris  is  intended  for  the  comparison 
of  a  series  of  observations  in  onU^r  to  determine  the  corrections  to  ho 
n|)|)lied  to  the  elements  hy  ;neans  of  the  differential  formula>  which 
we  shall  investij^te  in  tin.'  following  chapter,  it  will  always  he  ad- 
vis:il»ic  to  <'ompute  the  co-ordinates  hy  means  of  the  radius-vector 
and  true  anomaly,  since  both  of  these  quantities  will  he  reipiired  in 
limliiig  the  differential  coefficients. 

.'JH.  In  the  ease  of  a  hyiKrhoIic  orbit,  the  co-ordinates  may  be  com- 
puted directly  from  F,  since  we  have 


and,  eoiisequcntly, 


r  cos  i>  ^=a(e  —  pec  F), 
r  sin  V  — ■  a  tan  ■!  tun  F; 


T  —  (le  sin  a'  sin  A'  —  n  sec  Fsin  a'  sin  A'  +  «  tan  +  tan  Fsin  n'  cos  A', 
y  -^  uc  win  h'  sin  if  —  o  sec  Fm\  b'  sin  ii'  +  a  tan  4.  tan  i'sin  b'  cos  if', 
z  —  ae  sin  c'  sin  C  —  a  sec  i'sin  c'  sin  C"  +  «  tan  4-  tan  i'sin  c'  cos  C". 


Let  us  now  put 


Then  we  shall  have 


acsinrt'  sin  A'  =  X„ 

—  a  sin  a'  sin  A'  =^  fi,^, 
a  tan  +  sin  a'  cos  A'  ^=  v, ; 

nesin//  sin  if  =  ^j, 

—  «  sin  b'  sin  if  =  /ij, 
a  tan  4.  sin  b'  cos  K  ^^Vj\ 

aosinc'  sin  C"  =  A„ 

—  a  sin  c!  sin  C  =  iit, 
a  tan  4  sin  c'  cos  C  =  v,. 

X  =  A,  4-  /i,  sec  F  -f  ^'x  tan  F, 
y  =  kj-\-  /jty  sec  F  -\-  Vj  tan  F, 
8  =  ^,  -|-  /i,  sec  F  -\-v^  tan  i'. 


(106) 


In  a  similar  manner  we  may  derive  expressions  for  the  co-ordinates, 
in  the  case  of  a  hyjKsrbolic  orbit,  when  tlie  auxiliary  quantity  a  is 
1  used  instead  of  F. 

39.  If  we  denote  by  :r',  Q',  and  V  the  elements  wliich  determine 
the  position  of  the  orbit  in  space  when  referred  to  the  equator  as  the 


86 


TlIEOHETICAr.   ASTIIOXOMY 


fiiinliunciital  plaiu',  and  by  w.)  the  luiffiihir  distniicc  Ix'twccn  tlic 
iisc('ii(liii<f  nude  nf  the  orbit  on  the  iM-lifttic  and  its  asciMiding  n(Mlc  u\\ 
the  i'<|ualor,  \n-\uff  inca.snrcd  positively  from  the  t'«|uator  in  the 
direetion  of  the  motion,  we  sliall  liavc 

To  find  Si' and  /',  we  have,  fr<»m  the  spherieal  triangle  forme<l  hy 
the  intei-seftion  of"  the  planes  of  the  orbit,  ecliptic,  and  ecjuatur  with 

ihe  celestial  vault, 

cos  t'  =  cos  /  COS  s  —  sin  t  sin  £  cos  £1 , 
sin  (■'  sin  ^'  —-  sin  /  sin  JJ, 
sin  t'  cos  Q'  —  cos  i  sin  e  -f"  ^'"  '  ♦''os  «  cos  Q.  * 


Let  us  now  put 


nsin  iV=  cos?', 

n  co8iV=^  sin  i  cos  Ji, 


and  these  equations  retluce  to 


cos  «■'  =  «  sin  {N —  e), 
sin  i'  sin  Ji'  -  -  sin  /  sin  Ji, 
sin  i'  cos  ft'  =  «  cos  {2i  —  e)  ; 


from  which  we  find 

cot/ 


taniV: 


^      _,  cosiV 

tan  ft  =  — -i^    -    tan  ft, 

cos  (iV  —  e) 


cos  ft  ' 

cot  i'  =  tan  (iV—  e)  cos  ft' 


(107) 


Since  sin  /  is  always  positive,  cos  iV  and  cos  ft  must  have  tlie  same 
signs.     To  prove  the  numerical  calculation,  we  have 


sin }'  cos  ft 


cosiV 


sin  I  cos  ft'       cos  {N —  s) 

the  value  of  the  second  member  of  which  must  agree  with  that  used 
in  computing  ft'. 

In  order  to  find  ojp,  we  have,  from  the  same  triangle, 

sin  Wo  sin  ?"'  =  sin  ft  sin  e, 

cos  «»„  sin  i'  =  cos  £  sin  t  +  i^in  «  cos  i  cos  ft . 


Let  us  now  take 
and  we  obtain 


m  sin  M=  cos  t, 

m  cos  J!/=  sin  £  cos  ft  ; 


POSITION    IX   SPACE. 

cot  A/--    tuii£c;)sQ, 

cits  J/  _ 

cos  u»i  —  0 


97 


(108) 


nnd,  also,  to  check  the  calculation, 

sin  c  cos  Q  COS  3/ 

sin  i'  coH  w„       cos  {M  —  t)' 


If  we  apply  (lauss's  analogies  to  the  same  spherical  triangle,  we 


«et 


cof<.\/'  sin  .1  (  Si'  4-  "O  **'»  '  J^  '■'>■"  J  ('"  —  -). 
cos  }^i'  cos  J  (  ft '  ^-  % )  --■  cos  !  ft  cos  .1  ( /  +  e), 
ein  .W'  sin  '  (  ft'  —  mJ  -  sin  ;\ft  sin  !(»  —  £), 
ain  i*'  cos  i  ( ft'  ~  w„)  =  cos  ^ ft  sin  !  { /  +  t). 


(109) 


The  (puulrant  in  which  i(ft'  +  w„)  or  J  (ft  —  <«„)  is  situated,  must  Ik; 
so  taken  that  sin  J/' and  cos  J/' shall  l)e  positive;  and  the  agriM-incnt 
of  the  values  of  the  latter  two  (piantities,  computed  hy  means  of  the 
value  of  .J/'  derived  from  tan  J/',  will  serve  to  check  the  accuracy  of 
the  numerir.il  cahnilation. 

For  the  case  in  which  the  motion   is  repirdcd  as  rctr<»gra(le,  wc 
iiKist  use  180°  —  i  instead  of  /  in  these  eipiations,  and  we  have,  also. 


--ft +  ft' 


0' 


We  may  thus  find  the  elements  ;:',  ft',  and  !',  in  refei*cnce  to  the 
equiUor,  from  the  (ilemcnts  referred  to  the  ediptic ;  and  usinjr  the 
elements  so  found  instead  of  r,  ft,  and  /,  and  usini;  also  the  places 
of  the  sun  referred  to  the  ecpiator,  we  may  derive  the  heliocentric 
iuid  ••('ocentric  j)laces  with  respect  to  the  equator  hy  means  of  the 
iorniuia'  already  given  for  the  ecliptic  as  the  fundamental  plane. 

If  the  [Kisition  of  the  orbit  with  respect  to  the  ((juato'  is  given, 
and  its  position  in  reference  to  the  ecliptic  is  refpiinHl,  it  is  oidy 
luressary  to  interchange  ft  and  ft',  as  well  as  /  antl  180°  -  /',  s 
remaining  unchanged,  in  these  ecpuitions.  These  fornmlie  may 
alsd  he  used  to  determine  the  position  of  the  orbit  in  reterence  to 
any  plane  in  space;  but  the  longitude  ft  nuist  then  be  measured 
from  the  place  «)f  the  descending  node  of  this  plane  on  the  ecliptic. 
The  value  of  ft,  therefore,  which  must  be  used  in  the  solution  of  the 
Wjuations  is,  in  this  case,  eipial  to  the  longitude  of  the  ascending 
ntnle  of  the  orbit  on  the  ecliptic  diminished  by  the  longitude  of  the 
descending  node  of  the  new  |>lane  of  reference  on  the  ecliptic.  The 
qnaiitities  ft',  /',  and  to^  will  have  the  same  signification  in  reference 

7 


98 


TIIKOIIKTK  Ah   ASTItONOMY. 


tn  tilis  plniic  (Iiiit  tlifv  have  in  n'fcn'iicc  to  tlu*  <'f|iiat<)r,  with  this  dis- 
tinction, liowcvcr,  that  SI'  i'^  measured  from  the  (hx-entlin^  no(U' of 
this  new  |thine  of  relereiiee  on  the  eeliptie;  and  e  will  in  this  case 
denot«'  the  inclination  of  the  eeliptie  t(»  this  |)lanc. 


U).  We  have  now  derived  all  the  forninlip  which  can  be  reqnired 
in  the  ciise  of  nndistnrlx'd  motion,  for  the  eoni|intation  of  the  helio- 
centric or  jiciM'cntric  place  of  a  heavenly  ImmIv,  referred  either  to  the 
eclipti(  or  e«jnator,  <»r  to  any  other  known  plane,  when  the  elements 
of  its  orhit  arc  known  ;  and  the  formnliu  which  have  been  derived 
are  applicable  to  every  variety  of  conic;  section,  thus  including  all 
possible  Ibrms  of  undisturbed  orbits  consistent  v,ith  the  law  of  inii- 
versid  gravitation.  The  circle  i.s  an  ellipse  of  which  the  eccentricity 
is  zero,  anil,  conse(|nently,  M^=v=-u,  and  r  a,  for  every  point  of 
the  orbit.  There  is  no  instance  of  a  circular  orbit  yet  known  ;  but 
in  the  case  of  the  discovery  of  the  asteroi«l  planets  between  Mars 
and  Jupiter  it  is  sometiuies  thought  advisable,  in  order  to  tacilitate 
the  identili<'ation  of  comparison  stars  for  a  few  days  succeeding  the 
diseovcry,  toconipjite  circular  elements,  and  from  these  ar,  ephemeris, 

The  elenjcnts  which  determine  the  form  of  the  orbit  remain  con- 
stant so  Ic»ng  as  the  systi^m  of  elements  is  regarded  as  unchanged; 
but  those  which  determine  the  position  of  the  orbit  in  sjnice,  z,  Q,, 
and  /,  vary  from  one  j-poeh  to  another  on  aivount  of  the  change  of 
the  relative  p(jsition  of  the  planes  to  which  they  are  referred.  Tlnii: 
the  inclination  of  the  orbit  will  vary  slowly,  on  actc  :nt  of  the  change 
of  the  position  of  the  «'cHptie  in  space,  arising  from  the  perturbations 
of  the  earth  by  the  other  planets ;  while  the  longitude  of  the  peri- 
helion and  the  longitude  of  the  ascending  nmle  will  vary,  both  on 
account  of  this  change  of  the  position  of  the  plane  of  the  ecliptic, 
an<l  also  on  account  of  precession  and  nutation.  If  r,  Q,  and  /  an 
referred  to  the  true  equinox  and  ecliptic  of  any  date,  the  resulting 
heliocentric  places  will  be  referred  to  the  same  equinox  and  ecliptic; 
and,  further,  in  the  comi>utation  of  the  geot^jntric  places,  the  longi- 
tudes of  the  sun  must  be  referred  to  the  same  e(piinox,  so  that  the 
resulting  geocentric  longitudes  or  right  ascensions  Avill  also  be  re- 
ferred to  that  ecjuinox.  It  will  apj)ear,  therefore,  that,  on  acf;oiuit 
of  these  changes  in  the  values  of  r,  ft,  and  /,  the  auxiliaries  sin  a, 
sinb,  8\nc,  A,  B,  and  C,  introduced  into  the  formula;  for  the  co- 
ordinates, will  not  be  constants  in  the  computation  of  the  places  for 
a  series  of  dates,  unleys  the  elements  are  referred  constantly,  in  the 
calculation,  to  a  fixed  equin.x  and  ecliptic.     It  is  customary,  there- 


POSITION    IN   SPAfE. 


00 


fi>n',  (o  rt'»liu'('  tlic  elements  to  the  eeliptie  iiiul  mean  ef|nino.v  of  the 
lieifiniiin^  of  the  ynir  for  whieli  the  ephemeriH  is  required,  and  then 
to  compiite  the  phiees  of  the  planet  or  eoniet  referre<l  to  this  e<piinox, 
iwin^-,  in  the  ease  of  the  ri^ht  ase(>nsion  and  declination,  the  mean 
iililiipiity  of  the  ecliptic  for  the  date  of  the  tixed  e(piinox  adopted,  in 
the  computation  of  the  auxiliary  constants  and  of  the  eo-or«linates 
<»f  the  sun.  'I'lie  pli-'-es  thus  foinid  may  he  reduced  to  tlie  true 
(•<|uinox  of  the  date  l»y  the  well-Unown  formnhe  tor  precession  and 
mitatiou.  Thus,  for  the  rc(hu-tion  of  the  rij;ht  ascension  aiul  decliua- 
tjiin  from  tlie  mean  equiuox  and  eipiator  of  the  he^innin^  of  the 
year  to  the  apparent  or  true  eipiinox  and  cipiator  of  any  dat«',  usually 
the  date  to  which  the  co-ordinates  of  the  body  belong,  we  luive 


Att  =/  -|-  (f  sin  (  (i  +  a)  tan  »), 
A')  —:  (J  cos  (  G  -f-  o), 


(110) 


tlir  which  the  (piautities/,  7,  and  G  arc  derived  from  the  data  given 
cillicr  in  the  solar  and  lunar  tables,  or  in  astrouonucal  epheuu'rides, 
>iii(li  as  have  already  been  menti(Mied. 

The  problem  of  rc<luciiig  the  elements  from  the  ecliptic  of  one 
(late  /  to  that  of  another  date  V  may  be  solved  by  means  of  eipuitions 
(lo'.l),  making,  Ik-  •ever,  the  nwessary  distiiK-tion  in  regard  to  the 
point  from  whicii  Q,  ami  SJ'  are  measured.  JiCt  0  den(»te  the  longi- 
tude of  the  descending  node  of  the  ecliptic  of  /'  on  that  of  /,  and 
K't  7^  denote  the  angle  which  the  planes  of  the  two  ecliptics  make 
with  each  other,  then,  in  the  e(piations  (lOO),  instead  of  Q,  we  must 
write  Q,  —  d,  and,  in  order  that  i<,'  shall  be  measured  from  the 
Vernal  e<piiu(»x,  we  uuist  also  write  Q,'  —  0  in  place  of  Q,'.  Finally, 
we  uuist  write  !j  instead  of  e,  and  aw  for  10^,  which  is  the  variation 
in  the  value  of  (o  in  the  interval  V  —  /  on  account  of  the  ehanire  of 
the  position  of  the  ecli[)tic;  then  the  equations  become 


cos.Ji'  sin  I  iO,'  —  fl  +  Aw)  =  sin  \  (  Q 

cos  \l    cos  ]  (  ft'  —  ^'  +  Aw)  =:  COS  \  (  ft 

sin  Jj"  sin  {(Q,'  —  0  —  Aw)  =:_-.  sin  \  (  ft 
sin  ^t'  cos]  (ft'  —  tf  —  Aw)  ==  cos  1  Cft 


0)  cos  },  (i  — 15), 

0)  sin  Ui~rt),      ^■^^^'' 
0)  sin  \  (i  -|-  15). 


Tlu'so  equations  enable  us  to  determine  accurately  the  values  of  ft', 
1',  and  AW,  which  give  the  position  of  the  orbit  in  reference  to  th(» 
ooliptic  corresponding  to  the  time  t',  witen  6  and  jy  are  known.  The 
longitudes,  however,  will  still  be  referre<l  to  the  same  mean  equinox 
as  bofore,  which  we  suppose  to  be  that  of  t;  and,  in  order  to  refer 


100 


Til KOKKTKAIi   AHTUONO.M V, 


tlicm  to  the  moan  o(|uinox  of  tho  epoch  r,  l]w  nnionnt  of  tlic  pro- 
cession in  loni^itude  (hiring  the  interval  /'      /  niiist  a!.-*fi  be  upplie<l. 

If  the  changes  in  the  values  of  tlie  cleinent.s  are  not  ot  consi«h'r- 
ahh'  n)ai;nitti(h%  it  will  l)c  unno<'e,«Hary  to  iipply  these  rIpM'ou.s  forniuhe, 
and  we  may  derive  others  suniciently  exact,  and  puich  more  con- 
venient in  application.  Thus,  from  the  .spherical  triangle  formed  l>y 
the  intersection  of  the  plane  of  the  orhit  and  of  the  planes  of  the 
two  ecliptics  with  the  celestial  vault,  we  get 

sin  rj  cos  (SI  —  0)  ^^  —  cos  i'  sin  i  -\-  sin  i'  cos  i  cos  Aw, 

from  which  we  easily  derive 

sin  d"  —  t)  --  sin  >j  cos  (SI  —  <')  +  2  sin  i'  cos  i  sin'  ^Aw.        (112) 

We  have,  further, 

sin  Aw  sin  i'  =  sin  ij  sin  (Q  —  0), 


or 


.    ^  .      s\n(Sl—0) 

Sm  Aw  ■-=:  8U1  Tj .    'T, 


Hin  I 


We  have,  also,  from  the  same  triangle, 


which  gives 


sin  Aw  cos i'  =  —  cos  ( Ji  —  0)  sin  (SI'  —  0) 

+  sin  (SI  ~  0)  cos  (Si'  —  0)  COS)?, 


(li:0 


sin  (SI'  —  Sl)--=  —  sin  Aw  cosi'  —  2  sin  {SI  —  0)  cos  (ft'  —  0)  sin'  Jij, 


or 


8in(ft'  — ft)  =  —  sin )?  sin  (ft  — 0)  coti' 
~  2  sin  (ft  —  <^)co3(ft'— <^)  siuMij. 


Finally,  wo  have 


n'  —  7:=  ft'—  ft  +  Aw. 


(114) 


Since  r^  is  very  small,  those  equations  give,  if  we  ap])ly  also  the  pre- 
cession in  longitude  so  as  to  reduce  the  longitudes  to  the  mean  equinox 
of  the  date  /', 


Aw 


_    !>\n(fl--0) 


sm  I 


Aw' 


i'   =    i  -fj  COS  ( ft  —  0)  -|- 1 sin  2 1, 

Si'=  Si  +(it'-t)^^-71  Bm(Sl-0)coii'-}tsm2(Sl-0),      (115) 
^'  =   ::  +  (<'-0-f +  '?8in(ft-tf)tan^i'-|l'siu2(ft-<?)j 


POHITION    IN    bl'ACE. 


101 


ill  wliirii   — --   JH  thu  aniiiiiil   prcrcssioii   in   loii^ittnlo,  aiiil   in   wliicli 

H      2n(»2<}4".8.     In  most  rasoM,  tlio  lant  terms  (»f  tli»'  ^<.\•j^r('s^i(Ml.s  for 
/',  Q',  n\u\  z',  iM'injf  of  the  m'coikI  (trdcr,  may  In-  nc^jlcrfcj. 

Fur  t lie  case  in  which  the  motion  is  rc^anh-d  us  rctro^rnidr,  wo 
niii-'l  i»ut  1H()°  /' antl  IKO"  /',  iii'.f«'a(l  of  /  suul  /',  r(-i|M<»iv(|y,  in 
tlif  ((juationM  for  mo,  I',  and  JJ';  and  for  z',  in  this  case,  \vu  !i!ivc 

rr'— ffr^ft'-  ft   -A««, 
wliich  }^ivo9 

,1/  ,  1 

((t  *  '    M 

If  \\v.  adopt  Hossol's  dotrrmination  of  the  hini-sohir  prr.t'ssion  a!nl 
(if  the  variation  of  the  mean  ol)li(iuity  of  the  eelipti*-,  wc  have,  at  tin; 
time  1750  -f  r, 

dt 


=  50".21129  4-  0."00()2442%Gr, 
=   0".48892  —  0."00000r.l43r, 


and,  consequently, 

,  =  (0."48892  —  O."00000r,143r)  (f  —  t); 

and  in  the  computation  of  the  vahies  of  tliese  quantities  we  must  put 
r  --  i(/'+  0  —  1750,  (  and  /'  l)ein«;  expressed  in  yeai**". 

The  lonj^itude  of  the  (K'.seendin}^  node  of  tlie  ecliptic  of  the  time  t 
on  the  ecliptic  of  1750.0  is  also  found  to  be 

351°  3()'  10"  —  o".21  a  —  1750), 

which  is  measurctl  from  the  mean  equinox  of  the  iK'frinninfr  of  the  year 

1750. 

The  Itmgitude  of  the  descendiuf;  node  of  the  ecliptic  of  /'  on  that 

of  /,  measured  from  the  sanje  mean  cipiinox,  is  ccpial  to  this  value 

tliiiiiiiished  by  the  angular  distance  between  the  descending  ntHJe  of 

the  ecliptic  of  t  on  that  of  1750  and  the  descending  ucmIi^  of  the 

ecliptic  of  t'  on  that  of  t,  which  distance  is,  neglecting  terms  of  the 

siH'<ind  order, 

5".21(f-1750); 
and  the  result  ia 


or 


351°  36'  10"  —  5".21  (t  —  1750)  —  5".21  (f  —  1750), 
351°  36'  10"—  10".42(<  —  1750)  —  5".21  (<'  —  0- 


108 


iiii;<»itr.rM  Ai.  akiuonomy. 


To  I'rtliiiT  tlii"  litii^itiiilf  to  llif  111*1111  <'<|iiiiii)X  at  tin-  tiiiic  (,  \v«'  inuiit 
n<l(l  the  };iiu-i'ul  pHrc.^Hiiiti  iliiriii^  tli<>  iiitrrval  t       17'><i,  nr 

r>0".21(/-  IToO;, 
8u  tliat  \w  liiivc,  tinnlly, 

0  ~.  351  ^  .'.d'  10"  4-  :»»".7iK<  -  li.VM      .V'.'il  (t'-- 1). 

Wlicii  the  clcinciits  r,  52,  "H'l  '  Imvo  Im-oii  thus  rcdiUH-d  (Vom  the 
(M-liptif  atui  mean  <'i|iiiii<i\  to  wliiih  tlicv  arc  rclrrrc*!,  to  those  of  the 
(late  liir  wliieh  the  lielioecniric  or  M;(.(Meiitrie  |»hiee  is  retjiiireil,  tliey 
may  l)e  n>teri-ei|  to  tlie  apparent  eipiinox  of  the  <late  liy  applying  the 
nutation  in  h»ii^itii(h>.  'I'hen,  in  the  eiise  of  the  Jeterniinalion  ot'  the 
ri^ht  nHcension  and  (h'clinatioti,  usiuj^r  the  appat'ent  oMi(|uity  of  the 
eeliptie  in  the  eompiitation  of  the  eo-onlinates,  wi-  direetly  (»l»taiii  the 
phiee  of  th(>  hiKly  referred  to  the  apparent  e(piino\.  lint,  in  eoni- 
pntinn  a  series  ol"  plae»'>,  the  ehan^^es  whieli  tliiis  take  phiee  in  the 
dements  themselves  from  date  to  date  induce  correspond  in;;  chan^^es 
in  the  auxiliary  ipiantities  n,  l>^  r,  A,  II,  and  (\  so  that  these  are  no 
lonf^'or  to  ho  c(insi(h'red  as  constants,  hut  lus  eontinnally  elian^rin;;  tin'ir 
vahies  l>y  small  ditlcrences.  The  ditfen-ntial  formnlie  for  the  com- 
putation o*"  these  chancres,  which  are  easily  derived  from  the  eipiation> 
(!)!>),  \vill  i)c  ;;iven  in  the  next  chapter;  hut  they  are  perhaps  unncccs- 
sury,  since  it  is  generally  most  convenicni,  in  the  esiscs  which  occur,  to 
compute  tlu' auxiliaries  for  the  extreme  dates  for  which  the  ephemeris 
in  re(piired,  and  to  interpolate!  tlieir  values  for  internu<liate  dates. 

It  is  advisahle,  however,  to  reduce  ihe  clenuMits  to  the  ecliptic  and 
mean  ecpiinox  of  the  hci^innin^  of  the  year  for  which  the  ephemeris 
is  rcfpiired,  and  usinjr  the  mean  ohli(|uity  of  the  ecliptic  for  that 
epoch,  in  the  computation  of  the  auxiliary  constants  for  the  etpiator, 
the  resulting  jr,.(»centric  rijfht  ascensictns  and  declinations  will  he 
referred  to  the  same  etpiinox,  and  they  may  then  be  rodueod  to  the 
apparent  e(piinox  of  the  date  by  a[)plyin};  the  corrections  for  prwes- 
sion  and  nutation. 

The  places  which  thus  result  are  free  fmm  jnti'(il/(t.i'  und  ahrrrdflmi. 
In  comparinjjf  observations  with  an  ephemeris,  the  correction  for  par- 
allax is  .".pplied  directly  to  the  observed  apparent  places,  since  this 
correction  varies  for  ditlerent  places  on  the  earth's  surface.  The  <'nr- 
I'cction  ft)r  aberration  may  be  applied  in  two  ditlerent  motlcs.  W^- 
may  subtract  from  the  time  of  obstu'vation  the  time  in  which  tlic 
lif^ht  frouj  the  planet  or  comet  reaches  the  earth,  and  the  true  i)laci.' 
for  this  rtduci'd  time  i»  identical  with  the  apparent  place  for  the  time 


NI'MKKK  AI.    KXAMI'r,h>. 


103 


of  niMrviilioii ;  f»r,  In  caso  wv  know  tlic  daily  or  hourly  inolioii  of 
the  IkmIv  ill  i'i|;)it  a.M't'iisimi  dimI  dfcliiiatiuii,  \V(*  inity  coiuputc  \\u 
iiiiiiii>ii  tliii'iii^  the  interval  wiiit-li  is  n-(|uirf<l  tor  tlic  li^lit  to  pass 
t'roiii  the  IxmIv  to  tlic  eai'tli,  \vlii<'li,  Itciii;;  applied  to  tlu;  oltscrvcd 
|ihu'»',  jjivrs  tlu'  true  placf  for  tlu'  time  oC  ol>siM'vation. 

We  inav  also  include  tlic  alx-rnition  dircrtiv  in  the  cphcincris  hv 
ii-iiij:  the  time  /  WH'.lHj  in  compiitiii^  the  jfcocciitric  places  lor 
the  liiiic  /,  or  l»y  siihtractin^i  t'roin  the  place  tree  I'roiii  altcrration,  coiu- 
|iiiicd  lor  the  time  f,  the  motion  in  <i  and  o  dtirin^j^  the  interval 
|!)7'.7Hj.  ill  which  expression  J  is  the  distance  of  the  ImkIv  fntin  the 
earth,  and  U>7.7H  tin  nnmlM'r  of  seconds  in  which  lij^ht  traverses  the 
iiKaii  distance  of  the  earth  from  the  .siin. 

It  is  customary,  hr>wevcr,  to  compnte  the  t[ili'  meris  fn>c  from 
iilM'rralion  and  to  siihtract  the  (Imr  itj  n/x  rratiou,  4'.}1  .~HJ,  from  the 
time  of  ohservation  wlwii  <'omparin^  ol»servation«  "iili  an  ephemcris, 

;u rdiiii.r  to  the  first  method  ahov*'  mentimi'i,.     Tj.    phic.'s  of  the 

fill!  used  in  compntin<;  its  co-ordinates  must  also  he  five  from  aherra- 
tidii;  1:1  if  the  lonj;itndes  derived  from  the  sf^lar  tahlcs  inclntle 
nlM'rratioii,  tlu;  proper  correction  must  he  appl'cd,  in  order  to  obtain 
till'  trill-  lon}j;itnde  rc(|uired. 

II.  KxAMlM,i:s. — We  will  now  collect  together,  in  the  proper 
oiilrr  for  nnmcrical  calculation,  some  of  the  principal  formuhe  which 
liave  hceii  derived,  and  illustrate  them  hy  numerical  examples,  coin- 
im'iiciiijj  with  the  luse  of  an  elliptic  orhit.  Let  it  he  reijuircd  to  find 
the  geocentric  riffht  asc-ension  and  decliiiati(»n  of  the  planet  luiri/iininc 
0,  for  mean  midnight  at  Washinjrton,  for  the  date  iSIJo  Fehruary 
'J4,  the  elements  of  the  orhit  heing  as  follows: — 

Epoch  ■~=^  \W4  Jan.  1.0  (rreenwich  mean  time. 
i»/==      r  •_>!»'  40". 21 

[  hclii)tic  and  Mean 

ft  .^^20(i    42  4(t  .1:5   ■  ,,     !     ^.    ,^,.,  ., 

'"'  I  h^ouinox,  lo()4.0. 

i=      4    'Mi  .)()  ..-)1  j        ' 

V>^    11    1')  51  .02 

logrt  =  o.:wMi.'ny 

log/i  =  2.!)07«088 
H  =  U2«".o.574.5 

When  a  series  of  places  i.s  to  be  computed,  the  first  thing  to  he 
tloue  is  to  compute  the  auxiliary  constants  u.sed  hi  the  expression.s  for 
the  co-ordinates,  and  although  but  a  single  pl.uo  is  refjuired  in  the 
prohKin  propo.sed,  yet  we  will  proceed   in  this  manner,  in  order  to 


104 


TirEomrncA i.  ahtroxomy. 


«'xliil)it  tho  application  of  tlu>  rornmlu".  Since  the  elements  r,  Ji, 
and  /  are  referred  to  the  ecliptics  and  mean  e(|uinox  of  18(>4.(>,  we  will 
first  redu(!e  them  to  the  ecliptic  and  mean  e<iuinox  of  l8(>o.().  For 
this  reduction  wo  have  t       18()4.(),  and  ('      iHOo.O,  which  give 


dt 


—  rw»"  ', 


r)()".2:{9, 


0  --.-  :{r)2°  ">!'  41", 


r,  =  0".4882. 


Suhstituting  those  values  in  the  equations  (115),  we  ol)t;iin 

/'  —  /  r.:.  A/  ^  —  0".40,  A  Ji  =  -  -I-    -)\\".(y\  ,  ATT  ^.  -f  50".'23  ) 

and  hence  the  elements  whi<'h  determine  the  position  of  the  orbit  in 
reference  to  the  ecliptic  of  IHOo.O  are 


T=  44°  21'  2.r.;j2, 


ft       206°  4;r  3;5".74, 


4°36'50".ll. 


For  tile  siime  insiant  we  derive,  from  the  American  Ejilicuu'i'is  ttnd 

Noiiticdl  AliiKDHW,  the  value  of  the  mean  obliquity  of  the  ecliptic, 

which  is 

e  =  215°  27'  24".03. 

The  auxiliary  constants  for  the  equator  are  then  found  by  means  of 
tho  formulae 


cot  A  --- 
cotli- 
cot  C  = 
sina  — 


tan  p,  cos  /, 


„        tan  I 
tan  Eo  -         _ , 
cos  SI 


cos  i  cos  (Eq  -\-  e) 


tan  ft  cos  Eg         cos  e 

cost  sin(A^-j-0 

tan  ft  cos  E^  sin  e 

cos  ft  .    ,        sin  ft  cos  c 

-.        ,  8mfc=         .         — , 

sui  .4  sin  Ji 


sni  c 


sin  ft  sin  e 
sin  C 


The  angle  E,,  is  always  less  than  1S()°,  and  the  ((uadra:it  in  which  if  is 
to  be  taken,  is  indicated  directly  by  the  algebraic  sign  of  tan  /'7„.  The 
values  of  sin  a,  sin  />,  and  sin  c  are  always  positive,  and,  therefore,  the 
angles  A,  li,  and  ('must  be  so  taken,  with  resj)eet  to  the  (puulrant  in 
which  each  is  situated,  that  sin  A  and  cos  ft,  sin  Ji  and  sin  ft ,  and  also 
sin  ('and  sin  ft,  shall  have  the  same  signs.     From  these  we  derive 


A  =  2{>({°  ;?9'  r)".07, 

1^  =  205    5;-)  27  .14, 
C=-  212    32  17  .74, 


h)g:-inrt=r.  9.1H»!)7ir)(), 
log  sin  b  =^^  9.})7482o4, 
log  siuc^:  9.5222192. 


Finally,  the  calculation  of  these  constants  is  prove<l  by  means  of  the 
fornuda 


tan  t 


NUMEIUCAL   EXAMPLES, 
shift  m\e  m\(C  —  li) 


109 


sin  a  COS  ^l 


wliicli  fiivos  lofj  tjin  /  8.90()8S7o,  ivj^rceiu}];  with  the  vahio  8.9O08876 
(li  rived  directly  from  /. 

N(  \t,  to  find  ;•  iuid  u.  The  date  18(55  Febniarv  21.5  mean  time 
at  Washinjfton  reduced  to  the  meridian  of  (inH'nwicli  l)y  applyin;^ 
tlic  (litVerenct!  of  lon^rituih',  iV'  8'"  11".2,  becomes  iSfiT)  iM-hrnarv 
LM. 714018  mcjin  time  at  (JnH'nwich.  'Die  interval,  ihcri'tltre,  from 
the  epocii  for  which  the  mean  anomaly  is  ^iveii  and  the  date  lor 
wliicli  the  jjeocentric  place  is  retpiircd,  is  120.71  1018  days;  and  mul- 
tiplyin;r  the  mean  daily  motion,  {)28".r)r)74r),  hy  this  numher,  and 
iuldinj;  the  result  to  the  given  value  of  J/,  we  get  the  mean  anomaly 
for  the  reijuired  place,  or 

M^r  r  29'  40".21  +  108°  30'  57".14  =-  110°  0'  liVM. 

Tlio  eecentrie  anomaly  E  is  then  computed  hy  means  of  the  c({uation 

M=E— e  mi E, 

tlic  value  of  e  heinjif  expressed  in  seconds  of  arc.  For  Enri/nonie  wo 
liave  loijsinyr  logo  9. 2907754,  and  hence  the  value  of  e  ex- 
pressed in  seconds  is 

log  e  =  4.0052005. 

Uy  inean.s  of  the  equation  (54)  we  derive  an  approximate  value  of  E, 

nainciv, 

£„=119°  49' 24", 

tiio  value  of  r'  expressed  in  seconds  Wing  log  c^-  3.895976;  ami 
with  this  we  get 


M,  =  E,-e  sin  J5; :-  110°  0'  50". 


Then  we  have 


M~M„ 


'Mr. 


^  —  339".7, 


1— fcosA;  1.097 

Aviiieh  giv(s,  for  a  second  approximation  to  the  value  of  E^ 

.E„  =  119°4:V44".3. 
This  gives  JI/;-=  110°  0'  36".98,  and  henro 


A^.=  + 


0".37 
1.097 


+0".34. 


106  TIIEOKKTICAI-    AHTIfONOMY. 

Tlicrcfoiv,  '\-c  have,  for  a  tliinl  approximation  to  tlio  value  of  E, 

E=^\W  4;r  44".«54, 

Avhicli  rorjuircs  no  further  correction,  sine*'  it  satisfies  the  ecjuatioii 
between  M  and  A'. 

To  find  /•  and  r,  we  linvo 

\  r  sin  \v  =  V'ad  -\-  e)  sin  \E, 
Vr  cos  \v  ^=  l/a(l  —  e)  cos  \E. 

The   vahies  of  the   first    fiietors    in    the   second    n»end)ers  of  these 

equations  are:  \o\f\  (/(!+<■)  ^  0.*2:{28 104,  and  U)g  1  (7(1--'.')^^ 
0.1408741;  and  we  obtain 

V  =  129°  :r  .">0".r)2,  log  )•  =  0.4282854. 

Since  7:  —  ^=^  197°  37'  49".58,  Ave  have 

n  =  v-\--—Q,  =  ;]2()°  41'  40".10. 

The  heliocentric  co-ordinates  in  reference  to  the  equator  as  the  fun- 
damental plane  are  then  derived  from  the  e<[uations 

x  =  r  am  a  sin  (A  -f  n), 
y  ^=r  sin  />  sin  (  B  -f-  't^» 
2  =  r  sin  c  sin  (  C  +  «)» 

which  give,  for  Eiwynome, 

X  =  -  2.(5(51 1 270,  1/  =  +  0.3250277,  z  =:  +  0.01 1 048(5. 

The  Aiiwricnn  Nautical  Almanac  gives,  for  the  equatorial  co-ordi- 
nates of  the  sun  for  1865  February  24.5  mean  time  at  AN'ashingtoii, 
referred  to  the  mean  ecjuinox  and  equator  of  the  beginning  of  the 
year, 

A'==-f  0.9094557,  r=  —  0.3509298,  Z=  — 0.1561751. 

Finally,  the  geocentric  right  a.scension,  declination,  and  distance  are 
given  by  the  e(iuations 

y-\-Y  2  +  Z.  z  +  Z  ,      z  +  y^ 

tana  =  '    ,    -,,,        tan  o  = --—..sni  o —--,,, cos  a,         J=—     ,, 

.1-  +  A  y  +  J  x-\-A  sui ') 

the  fii'st  form  of  the  ctiuation  for  tan  J  being  used  when  sin  a  is 
greater  than  cos  a. 

The  value  of  J  must  always  be  positive;   and  d  cannot  exceed 
±  90°,  the  minus  sign  indicating  south  declination.   Thus,  we  obtain 


NUMERICAL   EXAMPLES.  107 

0      isr  8'  29".2{),        .J  =  —  4°  42'  21  "..")«;,        log  J  =.^  0.24r)00o4. 

To  reduce  a  niul  o  to  the  true  o<iiiinnx  and  ctjiiator  of  February 
i\.'),  wo  liave,  from  the  Ndutical  Ahnniidc, 

f=  4-  16".8(),  log«7  ==  l.OKW,  O  =  45°  16'; 

and,  substituting  tho.se  values  in  equations  (110),  the  result  is 

Aa  ^  -f  17".42,  A')  r.:  —  7",  17. 

Iltiicc  the  goooontrie  place,  relerrod  to  the  true  equinox  and  etjuator 
of  tlif  date,  is 


a  :^-  1><1°  «'  4(J".71,  5=-~4°  42'  28".73, 


log  J  .  -  0.24.")0n54. 


Wlifii  only  a  single  place  is  required,  it  is  a  little  more  expeditious 

t(i  ctiiupute  /•  from 

r  =  nil  —  e  vo9 E), 
ami  then  r  —  E  from 

sin  \(v  —  E )  --  V" sin  \<f  sni  E. 

Tims,  in  the  case  of  the  required  place  of  Enrynome,  wo  get 

log  r  ^  0.42828o2,  v  —  E^Si"  20'  b" m, 

t'  =  12U°  3'50".56, 

a'.'rcoiiig  with   the  values   previously  di'tormined.     The   calcidation 
[may  Ik'  proved  by  means  of  the  fonuula 


sin  ^ (v  -f-  E)  =  v-  cos  \<p  sin  E. 


[ill  the  ease  of  the  values  just  found,  we  have 

; .  r  4-  A' )  =  124°  23'  47".<50,  log  sin  J  (>-  -^  E)  ^  9.9160318, 

I  while  the  second  member  of  this  equation  gives 
log  sin  \  (V  +  E)  =  9.9165316. 

In  the  calculation  of  a  single  place,  it  is  also  very  little  shorter  to 

|i(iin|Miti'  tii*st  the  heliocentric  longitude  and  latitude  by  means  of  the 

(i|iiatinn.s  (82),  then  the  geixjcntric  latitude  and  longitude  by  means 

lot"  !8il)  or  (90),  and  finally  convert  those  into  right  ascension  and 

Idirjination  by  means  of  (92).     When  a  large  number  of  places  are 

Ix'  computed,  it  is  often  advantageous  to  compute  the  heliocentric 


If-- 


108 


TIIKORETKAI.   ASTRONOMY. 


co-ordiiintoH  directly  I'roin  the  eccentric  anomaly  by  means  of  the 
e<iuations  (lOo). 

The  calciihitiim  of  the  gecK-entric  |)Iace  in  reference  to  the  eeli|>iic 
is,  in  all  respects,  similar  to  that  in  which  the  e({iiator  is  taken  as  the 
fundamental  plane,  and  does  not  re(|iiire  any  further  illustration. 

'I'lie  determination  of  the  fjreoccntric  or  heliocentric  place  in  the 
cases  of  parabolic  and  hyperholic  motion  differs  from  the  process 
indicated  in  the  preceding  example  oidy  in  the  calculation  of  »•  and  r. 
To  illustrate  the  case  of  parabolic  motion,  let  < —  T'  75..'{G4  days; 
log  7       O.OGoOlHfi;  and  let  it  be  reijuired  to  find  r  and  i'. 

First,  wc  compute  m  from 


C 


»i 


0 


7- 


in  which  log  C^=---  9.9601277,  and  the  result  is 

Iogj»=:0.0125r,48. 


Then  we  find  31  from 
which  gives 


log3/=1.8S97187. 
From  this  value  of  log  M  wo  derive,  by  moans  of  Table  VI., 

V  =.  79°  55'  57".26. 

Finally,  )*  is  found  from 

(I 
r 

which  gives 


eos'.'.v' 


log  r==  0.1961120. 

For  the  case  of  hyperbolic  motion,  let  there  ha  given  t —  T- 
65.41236  days;  ^  =  37°  35' 0".0,  or  h.gc  =  0.1010188;  and  logd 
—  0.6020600,  to  find  /•  and  v.     First,  we  compute  iV^from 

in  which  log^  =  9.6377843,  and  we  obtain 

log  xV-.  8.7859356;  iV^=  0.06108514. 

The  value  of  F  must  now  be  found  from  the  equation 
iV  =  eA  tan  F  —  log  tun  (45°  +  ^  F). 


wlicrcni  H  = 


NUMKn'»:AT.    KXAMJ'LHS. 


10!) 


If  we  assumo  F-    30°,  u  jnore  appi'oxiniatc  value  may  ho  tlfrivcd 


from 


tan  F. 


N  ■{   lo^r  tan  (»()'' 


whi.-li  ^'ives  F,  ---  28°  40'  23",  an.l  licncc  .V,  -  0.072G78.  TIkmi  we 
(ompiitc  the  coiTection  to  Ix;  apjtiied  to  this  value  of  F,  hy  means  of 
till'  o<|iuition 


LF, 


(N-N,)coi^'F, 


s, 


?.U—v()hF,) 
whfTi'in  .s  ^  2062fi4".8;  and  the  result  is 

e^F,  -=  4.f)0J>7  ( iV  ~  X, ) «  =.  —  3°  3'  43".0. 
Hence,  for  a  seeoud  approximation  to  the  value  of  F,  wc  have 

F,  =  25"  30'  40".0. 
The  corresponding  value  of  iVis  A',  =  0.0G176o3,  and  hence 
aF,  =  r).l!M)(iV^—  N,)!>  =  —  12'  9".4. 

The  tliird  approximation,  therefore,  give.*;  F,  =^^25°  24' 30".6,  and, 
repeating  the  operation,  we  get 

/'=-25°24'27".74. 

\vlii(  li  requires  no  further  correction. 
To  find  r,  wc  have 

r  =  a  I         „  —  1  I , 
\  cos  F         J 

which  gives 

log  r  =  0.2008544. 

Then,  V  is  derived  from 


I  and  we  find 


tan  ^v  =  cot  54  tan  }^F, 
V  =  07°  3'  0".0. 


M'hcn  several  places  are  required,  it  is  convenient  to  compute  v 
and  /•  by  means  of  the  equations 


1/ 


r  sm  TiV  = 


V^nie  -\-  1) 

V^cos  F 
l^aie  —  1) 


sin  \F, 


Vr  cos  Av  =  ^—^'-1       —  cos  hF. 


110 


TIIEORETICAIi    ASTRONOMY. 


For  tho  {jivon  values  of  o  and  c  \\v.  have  log  1 ' a{c  +  1)  =^  0.4782049, 
logl  o{i'  —  1)  --^  0.01(X)821I,  and  hcncf  we  derivo 


V  =  (57°  2'  r){)".{>2, 


log}- =  0.2008545. 


It  remains  yet  to  illustrate  the  calculation  of  r  and  r  for  elliiJtio 
and  hyperbolic  orbits  in  which  the  eccentricity  differs  hut  little  t'ruu] 
unity.  First,  in  the  case  of  elliptic  motion,  let  t  --  T~  68.25  day*; 
c  —  0.9075212;  and  log  q  -^  9.7008134.     We  compute  M  from 


M={t-T)^y^^~, 


wherein  log  CJ,=  9.9601277,  which  gives 

log  3/^=2.1404550. 
With  this  as  argument  wc  get,  from  Table  YI., 

F-=  101°  38'  3".74, 
and  then  with  this  value  of  Fas  argument  wc  find,  from  Table  IX., 

A  ^  1 540".08,  B  =  9".506,  C  -^  0".002. 

1  e 

Then  wc  have  log  i  ==  ^og  r--—  =  8.217680,  and  from  the  equation 

V  =  F+  ^(lOOi)  +  5(100/)'+  C(lOOi)', 
we  get 

V  =  V+  42'  22".28  -f  25".90  +  0".28  =  102°  20'  52".20. 

The  value  of  r  is  then  found  from 

q(l-\-e) 
jt  — —  — t 

1  -\-  e  cos  v' 
namely, 

log  J- =  0.1014051. 

We  may  also  determine  r  and  v  by  means  of  Table  X.     Thus,  we 
first  compute  M  from 


M=  ^oStzU  .  v'7'o(l  +  9e) 


3 


B 


Assuming  ^  =  1,  we  get  log  M^=  2.13757,  and,  entering  Table  YI. 
with  this  as  argument,  we  find  ?<;=  101°  25'.  Then  we  compute  A 
from 


NUMERICAL   EXAMPLES. 


Ill 


which  f^ivos  A  ~  0.024985.     With  tliis  vahio  of  A  as  argument,  we 

fin<l,  from  Table  X., 

log  B  =  0.0000047. 

The  exact  value  of  M  is  then  found  to  be 

logJ/n=:2.137r)fi;}5, 
which,  by  means  of  Table  VI.,  gives 

w  =  101°  24'  36".26. 
By  means  of  this  wc  derive 

A  =  0.02497944, 
and  iience,  from  Table  X., 

log  C'=  0.0043771. 


Tiien  wc  have 
which  gives 


tan  iy-—  C'tan  hv 


^/5(l+e) 
\  Y+"  9^' 


V  =  102°  20'  52".20, 
agreeing  exactly  with  the  value  already  louud.    Finally,  r  is  given  by 


r  = 


from  which  we  get 


(1  -\-AC-')  cosMy' 
Iogr  =  0.1614052. 


Before  the  time  of  perihelion  passage,  t —  T  is  negative;  but  the 
vahie  of  v  is  computed  as  if  this  were  positive,  and  is  then  considered 
as  negative. 

In  the  case  of  hyperbolic  motion,  i  is  negative,  and,  with  this  dis- 
tiuetion,  the  process  when  Table  IX.  is  usetl  is  precisely  the  same 
as  for  elliptic  motion;  but  when  table  X.  is  used,  the  value  of  A 
must  be  found  from 


and  that  of  r  from 


^  =  (1^9!!*^""^"' 


r  = 


{I  — AC^)  cos' hv' 


the  values  of  log  5  and  log  C  being  taken  from  the  columns  of  th? 
table  which  belong  to  hyperbolic  motion. 
In  the  calculation  of  the  position  of  a  comet  in  space,  if  the  motion 


112 


THEORKTrCAL    ASTRONOMY. 


is  retr();fnul('  niul  tlic  inclinntion  is  rcj^urdod  as  loss  than  00°,  the  dis- 
tiiK'ti(tiis  iiulicatt'd  in  the  fbrninhu  must  he  carerully  noted. 

42.  M'hen  we  have  thus  e(>ni])uted  the  phu-es  of  a  planet  or  eoiint 
for  a  series  of  dates  e(|uidistant,  we  may  readily  interpolate  tl  e  plaees 
for  intennediate  dates  by  the  usual  forinuhe  for  interpolatioii.  Tliu 
interval  between  the  dates  for  which  the  direct  computation  is  made 
should  also  be  small  enough  to  permit  us  to  nof^leet  the  ett'eet  of  tlie 
fourth  ditferences  in  the  process  of  interpolation.  This,  how(!ver,  i.s 
not  absolutely  necessary,  provided  that  a  very  extended  series  of 
places  is  to  be  computed,  so  that  the  higher  orders  of  diflerences  may 
be  taken  into  account.  To  find  a  convenient  formula  for  this  inter- 
polation, let  us  denote  any  date,  or  argument  of  the  function,  by 
a  +  nto,  and  the  corresponding  value  of  the  co-ordinate,  or  of  tiie 
function,  for  which  the  interpolation  is  to  be  made,  by /'(a -|"  »«')• 
If  we  have  coiujiuted  the  values  of  the  function  for  the  dates,  or 
arguments,  <t  -  -  (o,  a,  a  +  lo,  a  -f-  2w,  Arc,  we  may  assume  that  an 
expression  for  the  function  which  exactly  satisfies  these  values  will 
also  give  the  exact  values  corresponding  to  any  intermediate  value 
of  the  argument.  If  we  regard  n  as  variable,  we  may  expand  the 
function  into  the  series 

fia  +  nto)  =/(a)  +  An  +  /?/i'  +  O/i'  -f  &c.  (116) 

and  if  we  regard  the  fourth  differences  as  vanishing,  it  is  only  neces- 
sary to  consider  terms  involving  n^  in  the  determination  of  the 
unknown  coefKcients  ^1,  />,  and  C.  If  we  put  n  successively  equal 
to  —  1,  0,  1,  and  2,  and  then  take  the  successive  differences  of  these 
values,  we  get 


J\a)  =/Ui) 


I.  Diff. 
-B+C 


II.  Dift:     III.  Diff. 

2B 

QC 


f{a  +  2u,)  ^/( « )-\-2A-]-4B-\-8C  ^'^'^^'^  '^ 

If  we  symbolize,  generally,  the  difference /(a  +  mo)  — f(a  +  (n  —  1)  o) 
by  /'  (<i  +  ( H  —  i)  <'i),  the  difference  /  (a  +  (n  +  i)  (o)  — /'  (a  +  (n  —  ^)  w) 
by /"(a  +  nio),  and  similarly  for  the  successive  orders  of  difTerences, 
these  may  be  arranged  as  follows : — 


Argument. 

Function. 

a  —  w 

/(«  — w) 

a 

fia) 

a-\-  la 

f(a  +  w) 

o  +  2w 

f{a  +  2<o) 

I.  Diff. 

/(a  +  W 
fia  +  ia,) 


II.  Diff. 


III.  Diff 


f'ia  +  a.)     r(«  +  -i-) 


INTEUPOLATION. 


113 


Comparing  these  expressions  for  the  ditlbrences  with  the  above,  we 


got 


c=5r(o+i«'). 


j?=i/'(«), 


A^f(a  +  -io,)  -  i/'  (a)  -  J/"  (o  -f  », 
which,  frojn  the  manner  in  whioli  the  ditlerencos  are  formed,  give 

C=  h  (f"  (a  4- ««)  -/"  (a)),  B  =  J/"  {n\ 

A  =f(a  +  a,)  -f(a)  -  -•/"  (a)  -  ^  (/"  (a  +  «/)  -/"  (a)). 

To  find  the  vahie  of  the  func^tion  corresponding  to  the  argument 
a  +  Jw,  we  have  n  --^  J,  and,  from  (110), 

/(a +  H  ==/(«) +  M  +  .|^  +  iC. 

8ul)s<tituting  in  this  the  values  of  A,  B,  and  C,  last  found,  and  re- 
ducing, we  get 

/(«  +  i'")  =  -i  (/(a  +  «")  +/(«))  -  i  ih  if"  («  +  «')  +/"  («^)). 

ill  which  only  fourth  differences  are  neglected,  and,  since  the  place 
of  the  argument  for  ?i  =  0  is  arbitrary,  we  have,  therefore,  generally, 

/(«  +  (n  +  A)  <«)  =  .',  (/(«  -^(n  +  l)o,)  -\-f(a  +  nuj)) 

-  h  (A  (/"  (a  +  in  +  1)  io)  +/"  (a  +  «.>))).  (117) 

Hence,  to  interpolate  the  value  of  the  function  corresponding  to  a 
(late  midway  between  two  dates,  or  values  of  the  argument,  for  which 
the  values  are  known,  we  take  the  arithmetical  mean  of  these  two 
known  values,  and  from  this  we  subtract  one-eighth  of  the  arith- 
iiK'tical  mean  of  the  second  differences  which  are  found  on  the  same 
iiorizontal  line  as  the  two  given  values  of  the  function. 

By  extending  the  analytical  process  here  indicated  so  as  to  include 
tlic  fourth  and  fifth  differences,  the  additional  term  to  be  added  to 
qiuition  (117)  is  found  to  be 

+  rh  (^  (f  («  +  (»*  +  1)  ">)  +/"  («  +  nio))), 

ami  the  correction  corresponding  to  this  being  applied,  only  sixth 
ditlbrences  will  be  neglected. 

It  is  customary  in  the  case  of  the  comets  \yhich  do  not  move  too 
rapidly,  to  adopt  an  interval  of  four  days,  and  in  the  case  of  the 
asteroid  planets,  either  four  or  eight  days,  between  the  dates  for  which 
the  direct  calculation  is  made.  Then,  by  interpolating,  in  the  case  of 
an  interval  w,  equal  to  four  days,  for  the  intermediate  dates,  we 
obtain  a  series  of  places  at  intervals  of  two  days;  and,  finally,  inter- 

8 


114 


Til K(>RETICAL   ASTRONOMY. 


polatiii;:;  for  tlio  dates  iiitcnncdiato  to  tli(>se,  wp  dorivo  the  places  at 
intervals  of  one  day.  AViien  a  scries  of  places  has  been  i-oinputed, 
the  use  of  dill'ereiices  will  serve  us  a  check  upon  the  accuracy  of  tliu 
calculution,  and  will  serve;  to  detect  at  once  the  place  which  is  not 
correct,  when  any  discrepancy  is  apparent.  The  jfrcatest  discordanct; 
will  Itc  shown  in  the  dillerences  on  the  same  horizontal  line  as  thu 
erroneous  value  of  the  function;  and  the  discovdance  will  he  jfreatir 
and  jfrcater  as  we  proceed  successively  to  take;  higher  orders  of  dif- 
ferences. In  order  to  provide  against  the  contingency  of  systematic 
error,  duplicate  calculation  should  be  nuide  of  those  quantities  in 
whi<'h  such  an  error  is  likely  to  occur. 

The  ephenieridcs  of  the  planets,  to  be  used  for  the  comparison  of 
observations,  are  usually  computed  for  a  period  of  a  few  wirks  before 
and  after  the  time  of  opjutsition  to  the  sun ;  and  the  time  of  the 
opposition  may  be  found  in  advance  of  tlie  calculation  of  the  entire 
ephcmeris.  Thus,  we  find  first  the  date  for  which  the  mean  longitude 
of  the  ])lanet  is  ccjual  to  the  longitude  of  the  sun  increased  by  180°; 
then  we  compute  the  etpiation  of  the  centre  at  this  time  by  means  of 
the  e([uation  (o.'>),  using,  in  most  cases,  only  the  first  term  of  the 

development,  or 

V  —  J/='2e8in3/, 

e  being  expressed  in  sec'ds.  Next,  regarding  this  value  as  con- 
stant, we  find  the  date  for  which 

L  +  equation  of  the  centre 

is  equal  to  the  longitude  of  the  sim  increased  by  180° ;  and  for  this 
date,  and  also  for  another  at  an  interval  of  a  few  days,  we  compute 
u,  and  hence  the  heliocentric  longitudes  by  means  of  the  equation 

tan  (/  —  Sl)  =  tan ii  cos  i. 

Let  these  longitudes  be  denoted  by  I  and  I',  the  times  to  which  they 
correspond  by  t  and  f,  and  the  longitudes  of  the  sun  for  the  same 
times  by  ©  and  O ' ;  then  for  the  time  t^,  for  which  the  heliocentric 
longitudes  of  the  planet  and  the  earth  are  the  same,  we  have 


or 


t,==t  + 


t,=t'-\- 


Z— 180° 


O 


CO'- 
F 


O)~C/'-0 
■180°— O' 


it'-t), 


(118) 


the  first  of  these  equations  being  used  when  I  — 180^ 


O  is  less  ■  ^f  ^^'^  P"t 


TIMK   OK   OPPOSIXrON. 


115 


tlinii  /'  ISO"  -  O'.  If  the  tim(>  /„  tliftors  ronsiilcmbly  from  t  or 
/',  it  may  1m'  ii('<'('ssMrv,  in  order  to  obtain  an  acciinitc  result,  to  repeat 
tlic  hitter  i)art  of  the  calculation,  usiM<;  /„  for  /,  and  taixiiijj:  f  at  a 
siiinll  interval  from  this,  and  so  that  the  true  time  of  opposition  shall 
fall  het\v(>en  /  and  /'.  The  lonj^itndes  of  the  pliiuet  and  of  the  sun 
must  1)(>  measured  from  the  sanii"  eipiinox. 

When  the  eeeentricity  is  considerahle,  it  will  facilitate  the  enleula- 
tidii  to  use  two  terms  of  eipiation  (o.''))  in  tin<lin<^  tlu'  etpiation  of  the 
(viitre,  and,  if  c  is  expressed  in  seconds,  this  gives 

v  —  M^  2c  sin  M  +  '\  ■  -  sin  2M, 

4    a 

H  Ixinj;  the  nunjher  of  seconds  correspond  inj;  to  a  lenj;th  <)f  are  e<pial 
to  tlie  radius,  or  2O02G4".8 ;  and  the  value  of  r  —  M  will  then  l)o 
('X|ir(ssed  in  seconds  of  arc.  In  all  cases  in  which  eircidar  arcs  are 
involved  in  an  ecpiation,  great  care  must  he  taken,  in  the  numerical 
a|i|>licution,  in  reference  to  the  homogeneity  of  th«>  different  t(>rms. 
If  the  ares  are  expressed  hy  an  abstract  nundier,  or  by  the  length  of 
arc  expressed  in  parts  of  the  radius  taken  as  the  unit,  to  express  thera 
in  seconds  we  must  multiply  hy  the  nund)er  20()2()4.8  ;  hut  if  the 
arcs  are  expressed  in  seconds,  each  term  of  the  e(juation  must  contain 
only  one  concrete  factor,  the  other  concrete  factors,  if  there  he  any, 
being  reduced  to  abstract  numbers  by  dividing  each  by  «  the  number 
of  seconds  in  an  arc  equal  to  the  radius. 

43.  It  is  unnecessary  to  illustrate  further  the  numerical  application 
of  the  various  formuhe  which  have  been  derived,  since  by  reference 
to  the  formula)  themselves  the  course  of  procedure  is  ob  us.  1 1 
may  be  remarked,  liowever,  that  in  many  cases  in  which  auxiliary 
angles  have  been  introduced  so  as  to  render  the  equations  convenient 
for  logarithmic  calculation,  by  the  use  of  tables  which  determine  the 
lojfurithms  of  the  sum  or  difference  of  two  numbers  when  the  loga- 
rithms of  these  nnmbei"s  are  given,  the  calculation  is  abbreviated, 
and  is  often  even  more  accurately  performetl  than  by  the  aid  of  the 
auxiliary  angles. 

The  logarithm  of  the  sum  of  two  numbers  may  be  found  by  means 
of  the  tables  of  common  logarithms.     Thus,  we  have 


If  we  put 


log(a  +  6)-loga(l  +  *-)=log6(l+^). 
log  tan  X  =  2  (log  6  —  log  a), 


no 


We  sliiill  Imvc 


or 


TIIKOIIKTUAL   ASTRONOMY. 

log  (a  -\-  b)  =  log  a  —  2  log  coh  r, 

log  (n  -\-  b)  —-■■  log  b  —  2  log  win  r. 


The  first  form  is  used  wlion  cosr  is  greater  than  sin. 7*,  and  the  scef)ntl 
form  when  cnsr  is  less  than  sin.r. 

It  should  also  l»e  ol»s('rve<l  that  in  the  solution  of  <'(|iiations  of  the 
form  of  (H!»),  after  tan  (P.-  ©) — using  the  notati(»n  of  this  particular 
wise — has  In-en  found  hy  dividing  the  second  equation  hy  the  first, 
the  second  mendtcrs  of  these  e(|uations  Ixjing  divided  l)yeos(^  —  O) 
and  sin  (A  —  ©),  resju'ctively,  give  two  values  of  J  eos,9,  which  should 
agrci'  within  the  limits  of  the  unavoidahle  errors  of  the  logarithmic 
tallies;  hut,  in  order  that  th(>  errors  of  these;  tahles  shall  have  the 
least  intluence,  the  value  derived  from  the  first  e(|Uation  is  t<»  he  pre- 
ferred when  cos(^.  —  ©)  is  greater  than  sin  (^  —  O),  and  that  derived 
from  tlu!  seeojul  equation  when  cos(^  —  ©)  is  less  than  sin  (A —  ©). 
The  value  of  J,  if  the  greatest  uecm'acy  possible  is  reipiired,  should 
bo  derived  from  J  cos  ,9  when  ^5  is  less  than  45°,  and  from  J  sin  ,9 
when  ,i  is  greater  than  4')°. 

In  the  application  of  munhers  to  equations  (109),  when  the  values 
of  the  second  members  have  been  computed,  we  first,  by  division, 
find  tanjift'  !  (o„)  and  tan  ^  (ft'  —  u)„);  then,  if  sin  A  (ft' +  w„)  is 
greater  than  cos  J  (ft' -f  <«„),  we  find  cos  J/' from  the  first  equation; 
but  if  sin  \  (ft'  +  <«„)  is  less  than  cos  J  (ft '  +  w,,),  we  find  cos  ^/'  from 
the  second  eciuution.  The  same  princi])lo  is  ai)plied  in  finding  am  It' 
by  means  of  the  third  and  fourth  ecpiations.  Finally,  from  sin  \i' 
and  cos  Jt'  wo  get  tan  U',  and  hence  t'.  The  check  obtiiincd  by  the 
agreement  of  the  values  of  sin  Ji'  and  cos  J/',  with  those  computed 
from  the  value  of  i'  derived  from  tan  Jt'^  does  not  absolutely  prove 
the  calculation.     This  proof,  however,  may  Ix;  obtained  by  means  of 

the  eciuation 

sin  «■'  sin  ft'  =  ain  i sin  ft, 
or  by 

sin  i'  sin  w„  =  sin  e  sin  ft . 

In  all  cases,  care  should  be  taken  in  determining  the  quadrant  in 
which  the  angles  sought  arc  situated,  the  criteria  for  which  are  fixed 
either  by  the  nature  of  the  problem  directly,  or  by  the  relation  of  the 
algebraic  signs  of  the  trigonometrical  iunctious  involved. 


DIFFEUENTIAL    FOUMLL.E. 


117 


ClIAPTKU     II. 


I>VF.«TIOATI()N  OF  TIIK  PFFFKKENTIAL  rORMULiK  WHlrll  EXPIIK'W  TMK  IIKI.ATIOM 
IIKTWKKN  Till'.  (iKOrKNTllK'  UH  IIKI.IOIKN  lUlC  ri.A(  t><  (>F  A  IIKAVh'.NLY  llODV 
AXK  THE  VAIIIATION    (IK   THE   ELEMENTS   OK   ITH  OUlllT. 

•It.  In  inaiiy  calculiitioiirt  n^latiiip;  to  the  motion  of  a  liciivcnly 
IxmIv,  it  becomes  neeessjirv  to  (letcrmiiic  tlie  variations  which  small 
increments  applied  to  tiie  values  of  the  elements  of  its  orbit  will  pro- 
duce in  its  jfcocentric  or  heliocentric  place.  The  torm,  however,  in 
which  the  j)rol)lem  most  frequently  presents  itself  is  t'lat  in  which 
:i|)pro.\iinate  elements  are  to  l)e  correctcil  by  means  of  tiie  (lilfercnccs 
b('twe(  n  the  [)laces  derived  from  computation  and  those  derivetl  from 
ul)servation.  In  this  cxse  it  is  rc(jnired  to  tind  '\(i  vnriatious  of  the 
clcnients  such  that  they  will  cause  the  dilferences  between  cahnilation 
and  oi)servation  to  vanish ;  and,  since  there  are  six  elements,  it  follows 
that  -  \  separate  ecpiations,  involvinj^  the  variations  of  the  elements 
a.s  the  unknown  (piantities,  must  be  formed.  F^ach  longitude  or  right 
asct.'usion,  and  eacli  latitude  or  declination,  derived  from  observation, 
will  furnish  one  e»|uation  ;  and  hence  at  least  three  (.'omplete  observa- 
tions will  be  required  for  the  solution  of  the  problem.  When  more 
than  three  observations  are  employed,  and  the  number  of  equations 
exceeds  the  number  of  unknown  quantities,  the  e(piations  of  condi- 
tion which  are  obtained  must  be  reduced  to  six  final  e(iuations,  from 
which,  by  elimination,  the.  corrections  to  be  applied  to  the  elements 
may  be  determined. 

If  we  suppose  the  corrections  which  must  be  ajipliod  to  the  ele- 
ments, in  order  to  satisfy  the  data  furnished  by  observation,  to  be  so 
small  that  their  squares  and  higher  powers  may  be  neglected,  the 
variations  of  those  elements  whi(;h  involve  angular  measure  being 
ox|>rossed  in  parts  of  the  radius  as  unity,  the  relations  sought  may 
be  determined  by  differentiating  the  various  formuhe  which  determitio 
the  position  of  the  body.  Thus,  if  we  represent  by  d  any  co-ordi- 
nate of  the  place  of  the  body  computed  from  the  assumed  elements 
of  the  orbit,  we  shall  have,  "i  the  case  of  an  elliptic  orbit. 


118 


THEORETICAL   ASTRONOMY. 


il/(,  being  the  moan  anomaly  at  the  epodi  T.  Let  d'  denote  the  vahie 
of  this  co-ordinate  as  derived  dircctlj'  or  indirectly  from  observation; 
then,  if  we  represent  the  variations  of  the  elements  by  Ci.ir,  aJJ,  a/, 
&c.,  and  if  we  suppose  these  variations  to  be  so  small  that  their 
squares  and  higher  powers  may  be  neglected,  we  shall  have 


do 


+  rfjr^^+ 


do 

dti. 


A.«. 


(1) 


The  differential  coefficients  -,— ,    ,_-,  &c.  must  now  be  derived  from 

d-    dSi 

the  equations  which  determine  the  place  of  the  body  when  the  ele- 
ments are  known. 

We  shall  first  take  the  equator  as  the  plane  to  which  the  positions 
of  the  body  are  referred,  and  find  the  diiferential  coefficients  of  the 
geocentric  right  ascension  and  dcqlination  with  respect  to  the  elements 
of  the  orbit,  these  elements  being  referred  to  the  ecliptic  as  the  fun- 
damental plune.  Ijet  x,  y,  z  be  the  heliocentric  co-ordinates  of  the 
body  ill  reference  to  the  equator,  and  we  have 


or 


Hence  we  obtain 


0  =f(x,  y,  z), 

do  =  -.-  dx  +  -,—  dy  -\-  -,—  dz. 
dx        '    dy    ^       dz 


dO^ 


do     dx 


do 


dx'  dr:        dy 


dy       do     dz 
d-K       dz     diT 


(2) 


and  similarly  for  tiiG  differential  coefficients  of  d  with  respect  to  tiie 
other  elements.  We  must,  therefore,  find  the  partial  differential  co- 
efficients of  d  with  respect  to  .r,  //,  and  z,  and  then  the  partial  differen- 
tial coefficients  of  these  co-ordinates  with  respect  to  *hc  elements.  In 
the  case  of  the  right  ascension  we  put  ^  =  a,  and  in  the  case  of  the 
declination  we  put  d  =  d. 

45.  If  we  differentiate  the  equations 

a; -|- JT  =  J  cos  ^  cos  o, 
y  -{-  Y=  J  cos  '5  sin  o, 
s  +  Z  =  J  sin  «J, 

regarding  X,  Y,  and  Z  as  constant,  we  find 


di 

'^.'o  find  the 

a;  =  r  cos  u 

y=:r  cog  u 

3  =  r  cos  u 

which  give, 

dx 

dSi 

<f!/ 

dQ 

DIFFERENTIAL   rORMUL.E. 


119 


dx  =  cos  a  COS  o  (JJ  —  J  sin  a  COS  '5  f/a  —  J  COS  a  sin  H  d<\ 
dij  =rr  sin  a  cos  S  dJ  -\-  J  cos  a  cos  o  da,  —  J  sin  a  siu  o  d<\ 
dz  =  sin  o  d  J  +  J  cos  d  dd. 

From  these  equations,  by  elimination,  wc  obtain 

,  ,             sin  a  ,     ,   cos  a  , 
cos  0  «a  =; j—  dx  -| .    dy, 


(3) 


,.           coso  sin')  ,        sin  a  sin  5  ,     ,   cos<5  , 
do  =  —  — — dx dy  +  — ,—  dz. 


Therefore,  the  partial  (liffercntial  coefficients  of  a  and  d  with  respect 
to  the  heliocentric  co-ordinates  are 


.  da  sin  a 

coso-y-  = 

dx  J 

,  da         COStt 

cos  0  -J-  =  0, 
dz 


cos  a  sin  (5 


dS  _ 

dx~ 

dj  _ 

dy-  J 

d(i  _  cos  5 

Iz  ~~  "j   ■ 


sm  a  sm ') 


(4) 


Next,  to  find  the  partial  differential  coefficients  of  the  co-ordinates 
X,  y,  z,  with  respect  to  the  elements,  if  we  differentiate  tlie  equations 
(lOO)i,  observing  that  sin  a,  sin  6,  sin  c,  A,  B,  C\  are  functions  of  £1 
and  i,  we  get 

dx  z::rz'  dr  -\-  X  cot  ( J.  +  vi)  du  -\-  -  '-  dQ  -\-    ,.  dl, 

dy  --=.  ^;  dv  -f -  y  cot  iB  -\-u)du-\-  ^'|  rfft  +  |(  di, 

dz  =z .  dr  4-  z  cot  {C -{-  u)  du  +  ,.-.  dSl  -\-  -rr  di. 
r  rtjj  (11 

ft  V         ft  1* 

'!.'o  find  the  expressions  for  -j^,  -j-,  &c.,  we  have  the  equations 

a;  =  r  cos  u  cos  ^  — r  sin  u  sin  SI  cos  i, 

y=:r  cos  n  sin  £1  cos  c  -{-  r  sin  u  cos  SI  cos  t  cos  s  —  r  sin  u  sin  i  sin  e, 

2  ==:  r  cos  u  sin  SI  sin  e  +  }•  sin  u  cos  $^  cos  i  sin  s  -j-  r  sin  it  siu  i  cos  e, 


which  give,  by  differentiation, 

dx 
dSi 

dy  _ 


dSl 


=  —  r  cos  M  sin  SI  — ''  sin  ^*  cos  JJ  cos  i, 

r  cos  tt  cos  Si  cos  e  —  r  sin  ii  sin  Ji  cos  i  cos  e, 


120 


dz 
dQ 
dx 
di 
dy 
di 


THEORETICAL  ASTRONOMY. 
-  :=  r  cos  u  cos  Q  sin  s  —  r  sin  u  sin  ^  cos  i  sin  e, 
r  sin  «  sin  Ji  sin  i, 
—  r  sin  M  cos  JJ  sin  i  cos  e  —  ?•  sin  u  cos  i  sin  e, 


- ;  .=  —  r  sm  «  cos  Q,  sin  i  sin  £  -\-  r  sin  it  cos  i  cos  e. 
The  first  three  of  these  equations  immediately  reduce  to 


do, 
and  since 


dx 

-J—  =  —  y  cos  £  —  z  sin  I 


dy 
do, 


X  cos  S, 


dz 
~d9, 


=  xsine;    (5) 


cos  rt  =  sin  5i  sin  i, 

cos  b  =^  —  cos  SI  sin  i  cos  e  —  cos  i  sin  e, 

cos  c  =  —  cos  JJ  sin  i  sin  e  -j-  cos  i  cos  e, 


we  have,  also, 
HI 


=  r  sin  it  cos  a, 


dy  .  ,  dz 

—V  =  ?•  sin  u  cos  0,  — , . 

ot  at 


»•  sm  tt  cos  c. 


du=^  dv  -\-  dTt  —  dSl, 


Further,  ^^e  have 
and  hence,  finally, 

X 

dx^=  -  dr  -\-  X  cot  {A  -\-  u)  dv  -\-  x  cot  (A  -\-  xC)  dn 

-\-  ( —  X  cot  (A  -f  u)  —  y  cos  s  —  2;  sin  e)  rfj^  -\-  r  sin  u  cos  a  rft,' 

dy  =  -dr  -\-  y  cot  (.B  -\-  «)  rfy  +  ^  cot  (J5  +  «)  <if 

+  ( —  y  cot  (-B  -f  0  +  ^  cos  e)  dSl  +  r  sin  u  cos  6  cf i, 

9 

t?2  =  -(ir  +  2  cot  ( C  -f-  ■'0  <?«  +  3  cot  ( C  +  «)  d:: 
-\-  ( —  2  cot  (  C  +  It)  +  «  sin  e)  dft  +  '"  sin  u  cos  c  di. 


These  equations  give,  for  the  partial  differential  coefficients  of  tho 
heliocentric  co-ordinates  with  respect  to  the  elements, 


(6) 


dx        dx  .r  A    t     \ 

dz  dz 

rfrr         dv 


*  =  ^  =  ,cot(B  +  ..). 


a  cot  (  C  -j-  u) ; 


DIFFERENTIAL   FORMULAE. 

(hj 


= — a;  cot  (-4  -}-  m) — j/  cos  e — z  sine, 


do, 


121 
=  —  y  cot(  B-\-  u) +x  COS  e, 


=  —  z  cot  ( C  +  It)  +  a;  sin  e ; 


_—  =  r  sin  «  cos  o, 

«( 

rf/ X 

dr       r 


dy  .  , 

— ,r  =  r  sm  u  cos  o, 
at 

dr       r 


dz  .  ,_^ 

-V-  =  r  sm  «  cos  c;  (7j 

rfr       r 


When  the  direct  inclination  is  greater  than  90°,  if  we  introduce  the 
(lis^tinction  of  retrograde  motion,  we  have 


and  hence 

dx  dx 


(/t 


dv 


du  =  dv  —  dTt-\-  dQ,, 
=  —  X  cot  (A  +  u), 


dy 


dy 
dv 


(fx       dx 


dz  dz  .rn  \      \ 

-^  =  -  —  :=-zcotiC+u); 

dij  __  d]i    , 


—  ycot{B  +  u), 
(8) 


dz         dz    ,        . 

,„  —  , 2/ cose — 3  sine,       -,-;:-= -;-t- a;  cose,      j7^=-i — h^sme. 

dQ,       dv      ^  dSl       dv  '       dSl       dv    ' 

Tiic  expressions  for  -7^>  ---,  and  -,'-  remain  unchanged;  and  we 


have,  also, 

dx 

-rr  ^=  —  r  sinitcosa, 

di 


dr     dr 


dr 


di 


J       dz 
■  —  rsinttcoso,    -t- 


•  r  sin  u  cos  c. 


(9) 


It  is  advisable,  in  order  to  avoid  tlie  use  of  two  sets  of  formula?,  in 
part,  to  regard  the  motion  as  direct  and  the  inclination  as  susceptible 
of  any  value  from  0°  to  180°.  If  the  elements  which  arc  given  are 
for  retrograde  motion,  we  take  the  supplement  of  i  instead  of  /;  and 
if  we  designate  the  longitude  of  the  perihelion,  when  the  motion  is 
considered  as  being  retrograde,  by  (j:),  wo  shall  have 

If  we  introduce,  as  one  of  the  elements  of  the  orbit,  the  distance 
of  the  perihelion  from  the  ascending  node,  we  have 


and,  hence, 


du  =  rfv  +  t?w, 


dx        dx  ^r  A    ,     \ 

-y-  =  -J-  =  X  cot  {A  4-  u), 
dm        dv 


dz_ 
du> 


dz 
dv 


dy  dy  

d(o        dv 

zcotiC-\-  u). 


y  cot(J5  +  «), 


(10) 


122 


THEORETICAL  ASTRONOMY. 


The  values  of    -->  -r'-,  and  --"  must,  in  this  case,  be  found  by  means 

of  the  equations  (5). 

By  means  of  these  expressions  for  the  differential  coefficients  of  the 
co-ordinates  x,  y,  z,  with  respect  to  the  various  elements,  and  those 
givciu  by  (4),  we  may  derive  the  differential  coefficients  of  the  geo- 
centric vight  ascension  and  declination  with  respect  to  the  elements 
SI,  i,  and  -  or  co,  and  also  with  respect  to  r  and  v,  by  writing  suc- 
cessively a  and  o  in  place  of  d,  and  ft,  /,  &c.,  in  place  of  ~  in  the 
equation  (2).  The  quantities  /•  and  r,  however,  are  functions  of  the 
remaining  elements  <f,  Mq,  and  /i;  and  we  have 


dr 


dr 


dr 


rf/^+?SA^'^^^o  +  i^'^' 


^'==Z'^'''^dJiL 


dv     ,,.    ,    dv   , 
dm.  -f-  - ,—  rf/jt. 
0  dli 


Therefore,  the  partial  differential  coefficients  of  x,  with  respect  to 
the  elements  f,  31^,  and  fx,  are 


dx 

d(p 

dx 

dM, 

jlx^ 

d[j. 


dx 
"cF 
dx 
Ir 
dx^ 
dr 


dr         dx 
d<p         dv 


dr 
d3L 


4- 


df 
dv 


dx^       _ 
dv  'dM^' 


(11) 


dr         dx 
d/j.         dv 


dv 

d;i  ' 


The  expressions  for  the  partial  differential  coefficients  in  the  case  of 
the  co-ordinates  y  and  z  are  of  precisely  the  same  form,  and  are  ob- 
tained by  writing,  successively,  y  and  s  in  place  of  x.     The  values  of 


dx 


dx 
dv 


dy 

17' 


dv 


dz 


dz 


dr 


,  and  -y-  are  given  by  the  equations  (7),  and 


when  the  expressions  for 


dv 

dr     dv      dr       dv     dr         ■,  dv  .,  , 

~r'  "V"'  ~rrF'  "ttt'  ~i~>  and  — r-  have  been 
d(p    dtp    dM^    dM^    d.'J.  d/x 

found,  the  partial  differential  coefficients  of  the  hel>'X!entric  co-oi'di- 

nates  with  respect  to  the  elements  <f,  31^,  and  /i  will  be  completely 

determined,  and   hence,   by   means  of  (2),   making  the   necessary 

changes,  the  differential  coefficients  of  a  and  d  with  respect  to  these 

elements. 


46.  If  we  differentiate  the  equation 

M^=E—eamE, 


DIFFEUEXTIAL   FORMULAE. 


123 


WO  shall  have 


dM^=  (IE (I  —  e  cos  E)  —  cos  ^  sin E  dip. 

But,  since  1  —  c  cos  J£  =  -,  and  cos  cp  sin  E=  -  sin  v,  this  reduces  to 
'  u  ^  a 

T  V 

dM=  -  dE  —  -  sin  v  d<p, 
a  a 


or 


dE  =  -  rfj/  -)-  sin  V  d<f. 


If  we  take  the  logarithms  of  both  members  of  the  equation 

tan  Iv  =  tan  A£tan  (45°  +  J,^), 
aud  (liflerentiate,  we  find 

dv  dE  ,  d(p 


+  ^ 


2  .<in  \c  cos  ly       2  sin  ^^E  cos  ^£  ^  2  sin  (45°  +  J  f  j  cos  (45°  +  ^y 

wliich  reduces  to 

,         sin  V   ,  _  ,   sin  v   , 

rfy  =  .     „dE  -\ dv. 

sui  E  cos  f 

Introducing  into  tliis  equation  the  value  of  dE,  already  found,  and 

V  8111  V 

replacing  sin  E  by ,  we  get 

*         *=  -^  acosv' 

a' cost"  ,,,  ,    sini'/rtcosV   ,   ^\  , 

dv  =  - — i—  dM-\-  ■ I ~  +  1    c?v'- 

1^  cosy\      r  I 

But  since  a  cos^<p  =p,  and  --  =  1  +  sin  <p  cos  v,  this  becomes 

,       a'  cosc>  ,,,  ,  /    2      .  ,  \  .      ■ 

^y  =: ^  dM  4- 1 [-  tan  f  cos  v  I  sm  v  dv. 

r'  ycosv  /  ^ 

If  we  diflferentuate  the  equation 

r  =  a(l  —  ecosE), 


(12) 


we  shall  have 


dr=  -da  -{-  ae  sin  JEJ  dE  —  a  cos  ^  cos  E  d<p ; 


a 


and  substituting  for  dE  its  value  in  terms  of  dM  aud  c?f ,  the  result 


IS 


dr=  -rfa-|-a  tau^  sinv  rfil/ +  (ae  sin  J?  sin  v — a  cos?  cos  J5J)t?^.     (IS) 
a 


124 


THEORETICAL  ASTRONOMY. 


„  .  .      r,        Sin  1)  COS  ^P  -  „         COSV  +  e  i     n  i 

J^ow,  since  8in^  =  v,    -       >  nntl  cos^  =  t— r >  wc  shall  have 

1  -\-e  cos  V  1  +  c  cos  u 


ac  sin  ^  sin  v  —  a  cos  99  cos  E  = 


ae  cos  v"  sin'  v       a  cos  v"  f  cos  c  +  c) 


1  -\-  e  cos  V 


1  +  e  cos  V 


which  reduces  to 

ae  sir.  E  sin  r  —  o  cos  ^  cos  £  =^-  —  a  cos  ^c  cos  v 
Hence,  the  expression  for  dr  becomes 


dr=  -  da  -\-  a  tan  ^  sin  v  dM  —  a  cos  f  cos  v  (?^. 
a 

Further,  we  have 

T  being  the  epoch  for  M'hich  the  mean  anomaly  is  Mq,  and 

kVl  +  m 


(14) 


M  = 


ai 


Differentiating  these  expressions,  we  get 

dM=  dM,  +  {t  —  T)  dfi, 
da  i     ^1^  , 


a 


''   f^' 


and  substituting  these  values  in  the  expressions  for  dr  and  dv,  we 
have,  finally, 

dr  =  a  tan  ^  sin  v  dM^  -j- 1  a  tan  ^  sin  v(t  —  T)  —  —  \dii 

—  a  cos  tp  cos  V  dtp,  (15) 

,        a' cos  V  ,,,,«' cos  ^  ,.       mx  7     ,  /     2      ,  ^  \  .       J 

dv  = 7-^  dMa  +        ,      (^  —  T)  f?M  + -f  tan  <»  cos  v   sm  v  rf^'. 

r*  "  r^  \cos^  / 

From  these  equations  for  dr  and  dv  we  obtain  the  following  values 
of  the  partial  differential  coefficients : — 


dr 

df 

dr 

dFL 


=  —  a  cos  If  cos  V, 
=:  a  tan  v  sin  v, 


,cos^ 

rfv  a'cos^ 

dTL 


^a      ' 


tan  p  COS  r  isinr, 

(16) 


^=atan^sin. («-r)-|r  206264.8,  ^  =  ^«-n 
a/t  op.  dfi  r 


DIFFERENTIAL    FORMULJE. 


126 


It  will  be  observed  that  in  the  last  term  of  the  exi>res.sion  for    ,    we 

juivo  8ui)])ose(l  n  to  be  expressod  in  scco/ids  of  are,  and  hence  the 
factor  20«j2()4.8  is  introduced  in  order  to  render  the  equation  homo- 
geneous. 

47.  The  formula)  already  derived  are  .sufficient  to  find  the  varia- 
tions of  the  right  ascension  and  declination  corresponding  to  the 
variations  of  the  elements  in  the  case  of  the  elliptic  orbit  of  a  planet; 
l)iit  in  the  case  of  ellipses  of  great  eccentricity,  and  also  in  the  cases 
of  parabolic  and  hyperbolic  motion,  these  formuhe  for  the  differential 
coetHcients  require  some  modification,  which  we  now  proceed  to 
develop. 

First,  then,  in  the  case  of  parabolic  motion,  sin  ^  =^  1,  and  instead 
of  J/j,  and  n  we  shall  introduce  the  elements  T  and  q,  the  differential 
coeflicients  relating  to  ;t,  ^,  and  i  iiaining  unchanged  from  their 
form  as  already  derived. 

U  we  differentiate  the  equation 

^^tzp.  _  g!  (tan  {V  +  \  tan*  {v), 
regarding  T,  5,  and  v  as  variable,  we  shall  have 

-7=  =  3 TT—  da  +  ■'i'?^  sec* iv, 

or,  since  7*^  =  (f  sec*  \v^ 


MT 


~ 7^—  f^'Z  +  2  "i  dv. 


IV2 


Q' 


2,^ 


Multiplying  through  by  — -.  and  reducing,  we  get 


dv  =  — 


(17) 


Instead  of  q,  we  may  use  log  5,  and  the  equation  will,  therefore, 

oecome 


2rU„ 


log?. 


(18) 


in  which  ?,q  is  the  modulus  of  the  system  of  logarithms. 


126  TIIRORETICAL   ASTRONOMY. 

If  WO  take  the  logarithms  of  both  members  of  the  e(iiiation 


and  (lifforentiate,  wo  find 


cos' .,  V 


dr  =  -  dq  +  r  tan  \v  dv. 
Introducing  into  this  equation  the  value  of  dv  from  (17),  we  get 
1       U  (t—T)  tan  ^ I'  \  ,        k  V  2q  tan  \v 


dr 


dT.        (19) 


Now,  since  — '  ■■  =  q  (tan  ^v  +  itan''  Iv),  and  q  =  r  cos^  ^v,  wc  have 

V  2q 

1       2k  (t  —  T)  tan  ^v  _1 


rW'lq 


(1  4-  tan'  \v  —  3  sin'  Av  —  sin'  Av  tan'  \v) 


1 
cos  y 


Wc  also  have 


k  V'lq        ,    kV  2q  cos'  U'  tan  iv ^sin  v 

Therefore,  equation  (19)  reduces  to 

^sinv 
dr  =  cos v «fl ,r_z-  ft i. 

If  we  introduce  d  log  (/  instead  of  dq,  this  equation  becomes 

k  sin  v 


(20) 


,            OCOSV    ,,                    A-SMIV      ^ 
dr  =  -   -r a  log  O T-znr  dT. 

^0  ^^        V2q 


(21) 


From  the  equations  (17),  (18),  (20),  and  (21),  we  derive 

dv       _  _kV'2q 
df     ~  /'~' 

rfy       __U:(t  —  T) 

dq      ~         '•V27 


dr 

^-si 

df      ~ 

~      V" 

dr 
dq 

=  COS  V, 

Jir_ 
rf  log  0  ~ 

qco^v 

_ ',    m 


dv   _  _  3A(«--jr)  v^. 

dlogq~ 


2A,r' 


and  then  we  have,  for  the  differential  coefficients  of  x  with  respect  to 
T  and  q  or  log  q, 


DIFFERENTIAL    FOIlMULiE. 


127 


dx       dx     dr       dx     dv                 dx 
dT~  dr'dT^  dc'df                d,i~ 

dx     dr       dx     dv 
dr     dq         dv  '  dq' 

dx          dx        dr       ^    dx 
dlogq       dr  '  dlogq       dr 

dv 
'dlogq' 

and  similarly  for  the  ditrorential  cot'tHcionts  of^  //  and  z  with  respect 
to  these  elements.  The  ex|)ressioiis  for  the  partial  (liffereiitial  eo- 
otHcieiits  of  X,  y,  and  2,  res[)eetively,  with  resi)e<'t  to  r  and  v  are  the 
■same  as  already  found  in  the  ease  of  elliptie  motion.  AV(t  shall  thns 
ohtnin  the  equations  whieh  express  the  relation  between  the  variations 
of  the  geoeentrie  ])Iaees  of  a  eoniet  and  tlu;  variation  of  the  parabolic 
elements  of  its  orbit,  and  which  may  be  employed  cither  to  correct 
the  approxiniate  elements  by  means  of  e<[uations  of  condition  fur- 
nished by  comparison  of  the  computed  j)lace  with  the  observed  place, 
or  to  determine  the  change  in  the  geocentric  I'ight  ascension  and 
(Iwliiiatiou  corresponding  to  given  increments  assigned  to  the  ele- 
ments. 

4(S.  We  may  also,  in  the  case  of  an  elliptic  orbit,  introduce  T,  q, 

and  e  instead  of  the  elements  (p,  M^,  and  [t.     If  we  differentiate  the 

expression 

5  =  a  (1  —  e), 
we  shall  have 


da 


We  have,  also, 


-dq  -{-  -de, 
q  q 


in  which  Tis  the  time  of  perihelion  passage,  and 

dM=  —  ifcV^r+m  a- 1  dT—  pv^l  +  m  ar'i{t—  T)  da, 
Plenee  we  derive 


kVl  -f-  m  a~  i 


kvl  +  mfrs 


(« —  T)  de. 


(t—T)dq 


Suhstituting  this  value  of  d3I  in  equation  (12),  replacing  sin^  by  e, 
and  reducing,  we  get 


dv  = 


=  -^J^lPSJL±^dT--i^-^^^kt-T)dq 


qr 


_^^jtV,-a+.n)„_r)_(g  +  i)„„,)_J_,,.        (23) 


128 


TIIKOnETICAL   ASTUOXOMY. 


In  a  similar  manner,  hv  snhstituting  the  values  of  da  and  (DI  in 
equation  (14),  and  reducing,  we  find 


dr  =-- 


-^- — e  emv  dT 


kl^l-\-m(t—  T) 


+  I />(**  —  cos vj—  Ul/pil  +m)  a 


^-_|_e8in.)r/7 


T) 


esin 


v\     1 
)l  — e^ 


de.   (24) 


These  e(jnations,  (23)  and  (24),  will  furnish  the  expressions  for  the 

^.  ,    ,,„.       ^.  ,        «...     dv    dv    dv    dr    dr        ,  dr      ,  .  , 
partial  (litierential  coeihcients  777,. -:->  ^-»  TTi,''-^''  Jvntl  -:-»  wluch  arc 
'■  dT   dq    de    dT  dq  de 

required  in  finding  the  differential  coefficients  of  the  heliocentric  co- 
ordinates with  respect  to  the  elements  T,  q,  anti  e,  these  quantities 
being  substituted  for  iJ/„,  n,  and  f,  respectively,  in  the  equations  (11). 

49.  When  the  orbit  is  a  hyperbola,  wc  introduce,  in  place  of  J!/„, 
fi,  and  (f,  the  elements  T,  <],  and  a^. 
If  we  diflerentiate  the  equation 


we  shall  have 


iVo  =  e  tan  F  —  log,  tan  (45°  +  i^F), 


dN„ 


[cosF      ^j 


dF 

cosF 


^  -|-  tan  i^  de, 


which  is  easily  transformed  into 


,-_      r     dF     ,   ,      „  tan4   ,, 

dK  = f^  +  tan  F- — --  d-^, 

"      a    cosii'   '  COS+       ' 


or 


dF 
sin  F 


a        ,„       a    tan^< 
r  tan  F      "      r     cos  ^z 


Let  us  now  take  the  logarithms  of  both  members  of  the  equation 

tan  -\F=  tan  ^v  tan  l^*, 
and  differentiate,  and  we  shall  have 


dv  =  sin  t) 


smv 


dF 


sin  F       sin  •4- 


dF 


Introducing  into  this  equation  the  value  of  — . — ^  already  found,  we 
get 


,         asm'y   ,,-.      lasmv 
r  tan  i<      "      \     r 


taniV        sm 


COS  ■^.        sin  4 


Ih- 


DIFFERENTIAL   FORMULA.  J  29 

But,  slncn  r  sin  v  =  a  tan  i//  tan  F,  and  p  ^-  a  tan^  \//,  this  reduces  to 


If  wc  differentiate  the  equation 

\  cos  F         I 


(25) 


wo  get 

dr  =  -  rfa  +  ae  tan'  F-~.  -^  -\ ^^  •  rt+. 

a  sniF       coaF     cos  + 

Substituting  in  this  equation  the  value  of  ~^~iriy  we  obtain 


,       r  ,     .  a^e  tan  F  ,  -^ 

dr  =  -  rfa  +  ■ (<iv„ 

a  r 


;-(- 


'e  tan"  F 


osFJ 


tan  4 


coax'  /  cos 4- 

which  is  easily  reduced  to 

,        r  J     ,       sin V    ,,^    ,  pi     r  ae      ,       \ 

a       '      sm^-       "      r\cosF      cos' F^      } 
Bat,  since 


d^, 


d^ 
sin  V 


ae 


a 


this  reduces  to 


cos  F      cos'  F      cos  F 


dr^'-da  +  '^^dN.  +  P^ie ^      )4 

o  sui  4-       "        r  \         cos  i'  /  81 


f/4, 
sin  4-* 


or 


a  sm4        "      ^   sin4< 

Now,  since  5  ==  a(c  —  1),  we  have 

,        q  .     ,   a  tan  4-  , 
^       a  cos  4/ 


(26) 


or 


Wc  have,  also, 
and  hence 


da  =  -  dq d^. 

q    ^       ^cos^ 


N,  =  ka-i{t—T), 

dNo  =  —  kcrUT—  ^ka-^it  —  T)  da. 
By  substituting  the  value  of  da^  this  becomes 

■JJT.  =  -  ka-idT-  VS€ll!^zIl  i,  +  m^ml 
"  q  ^  ag  cos  4. 


a^. 


130 


'niKOItKTTCAI,    ASTUOVOMY. 


Suhstitntinj^  this  value  of  ilj\\,  in  ('(|uatioii  {2^)),  and  reducing,  \vo 
ubtuin 


dv  = 


Wi 


dT- 


\kVpit-  T) 


q," 


dq 


+  (3'v.'(«-r)_/,      v.,  w/. 

\  qr  \r  }  hm^ 

In  n  similar  intmncr,  substituting  in  equation  (2G)  the  values  of 
da  and  f/A„,  and  reducing,  we  get 

,  k        sinv    ,rr  ,  l^       ,k(t  —  T)  sinv         \  , 

dr  = •  -  dT-\-\  -  —  3 --,—  •  — )dq 

V  p     cosi  \<1  V2q^       cos  J  4- J 'cos  4/ 

,  lUVpit—  T)     sinv       /»•  \    \  d\ 

a.  I  -J       ' - ..  —  cos  V  I/)  I   .      . 

\  fl  008  4-       \7  /'  /sin 4 


(28) 


The  equations  (27)  and  (28)  will  furnish  the  expressions  for  the 
partial  dill'crential  cocHlcients  of  r  and  c  with  respect  to  the  elements 
T,  7,  and  ^,  reipiired  in  forming  the  equations  for  coAthla  and  dd. 
It  will  be  observed  that  these  equations  are  analogous  to  the  equa- 
tions (23)  and  (24),  and  that  by  introducing  the  relation  between  e 
and  ■\^,  and  neglecting  the  mass,  they  become  identical  with  them. 
We  might,  indeed,  have  derived  the  equations  (27)  and  (28)  directly 
from  (2."})  and  (24)  by  substituting  for  e  its  value  in  terms  of  \//;  but 
the  (liH'erential  formula)  which  have  resulted  in  deriving  them  directly 
from  the  equations  for  hyperbolic  motion,  will  not  be  superfluous. 

50.  It  is  evident,  from  an  inspection  of  the  terms  of  equations  (23), 
(24),  (2"),  and  (28),  which  contain  de  and  </\^,  that  when  the  value  of 
e  is  very  nearly  e(jual  to  unity,  the  coefficients  for  these  differentials 
become  indeterminate.  It  becomes  necessary,  therefore,  to  develop 
the  corresponding  expressions  for  the  ease  in  which  these  equations 
are  insuflicient.     For  this  pur*  ose,  let  ua  resume  the  equation 


k(t-T)(l-\-e)^ 
2g* 


=  M-t  it* 


in  which  u  =  tan  ^v,  and  i 


2iiiu'  +  in«)  +  Si^iiv^  +  4u')  -  &c., 


Then,  since 


1  +e 
i=i(l_e)  +  |(l_e)«  +  &c., 

Vi^-e  =  Vr=r.r^rTr:;)  =  1  +  Ki  - «)  +  .^  (1  -  e)' +  (fee.. 

we  shall  have 


*ce_^jr) 

V2qi 

+  (A' 

If  it  is  required  to 

V!iriati(m  of  the  eleme 
small,  we  may  regard 
terms  multiitlicd  by  t 
(litVerentiating  the  ecpi 
regarding  u  and  r  as  v 

0  =  (1 
and,  since  da  —  i(l  + 


The  values  of  the  seee 
of  t',  may  be  tabulated 

is  by  no  means  indis] 
chanired  to  another  fo 
same  facility.     Thus,  I 


and  since,  in  the  case  c 

k(t—T)_ 
V  2  52 

this  becomes 

dv  _ 
de 

If  we  differentiate  \ 


regarding  ?•,  v,  and  e  t 

dr^ 
de 


DIFFERENTIA r.    FORMTL.E. 


131 


k(t  -  T) 


■«  +  !''<'+(.!« 


i«»)(,l-c) 


V2ql 

+  f /a«  -  ■li"''  +  A"';  (1  -  «)'  +  At-. 


(29) 


If  it  is  required  to  find  tho  cxprossioii  for    .-  in  the  ease  of  the 

variation  of  tho  dements  of  parabolic  motion,  or  when  1  — e  in  very 
small,  we  may  rej^ard  the  eoetlieient  of  1  —  e  as  constant,  and  nej^lect 
terms  multiplied  l)y  tho  s((uaro  and  higher  powers  of  1  — c.  By 
dillerentiating  the  e(piation  {'2U)  according  to  these  conditions,  and 
regarding  u  and  c.  as  variable,  we  get 

0  =  (1  +  u')  (In  —  (I It  ~lu*—  lu^)  de; 

and,  since  du  —  J(l  +  u^)  dv,  this  gives 


dv 
Ik 


}.n 


'.It' 


iW 


(1  +  u^r 


(30) 


Tho  values  of  the  second  member,  corresponding  to  different  values 
of  V,  may  be  tabulated  with  the  argument  v;  but  a  table  of  this  kind 

is  by  no  means  indispensable,  since  the  expression  for  -j-  may  bo 

changed  to  another  form  which  furnishes  a  direct  solution  with  tho 
same  facility.     Thus,  by  division,  we  have 

and  since,  in  the  case  of  parabolic  motion, 

k{t—T) 


V2ql 


=  «  +  W, 


r^  =  5»  (1  -t-  u»y, 


this  becomes 


^'"  _  9 


^('-^)./^_|taui. 


l/2r 


(31) 


If  we  differentiate  the  equation 

1  -\-  e  cos  V 

regarding  r,  v,  and  e  as  variables,  we  shall  have 

dr 2r"  sin*  jV   ,    r'e  sin  v     dv 

le  ~  qCl  +  e)'  "^  3 (1  +  e)  '  de " 


(32) 


132 


THEORETICAL  ASTRONOMY. 


In  the  case  of  parabolic  motion,  e  =  l,  and  this  equation  is  easily 
translbinaed  into 


dr 

de 


—  1 


ir  tan 


>(taM»  +  2^). 


(33) 


dv 


Substituting  for  -.—  its  value  from  (31),  and  reducing,  we  get 


dr 
de 


g  h{i—T)  .       ,    ,    ,    , , 


V2q 


(34) 


dv         dv 
The  equations  (31)  and  (34)  fuinish  the  values  of  -j-  and  -j-  to  be 

used  in  forming  the  expressions  for  the  variation  of  the  place  of  the 
body  when  the  parabolic  eccentricity  is  changed  to  the  value  1  +  de. 

When  the  eccentricity  to  which  the  increment  is  assigned  differs  but 

dv 
little  from  unity,  we  may  compute  the  value  of  --  directly  from 

(XC 

equation  (30).     A  still  closer  approximation  would  be  obtained  by 

dv 
using  an  additional  term  of  (29)  in  finding  the  expression  for  j- ;  but 

a  more  convenient  formula  ma}'  be  derived,  of  which  the  numerical 
application  is  facilitated  by  the  use  of  Table  IX.  Thus,  if  we  difter- 
cntiate  the  eqrucion 

v^V+A  (1000  +  B  {lOOiy  +  C(lOOi)', 

regarding  the  coefficients  A,  B,  and  C  as  constant,  and  introducing 
the  value  of  i  in  terms  of  c,  we  have 


de 


dV 
de  ' 


2Q0A  4005    ,,„..,  600  C     ,,.„.,, 

r,  (loot)  —  -pf-j^^,  (looiy, 


s(l-fe)-      Hl  +  e) 


s{l-\-er 


in  which  s  —  206264.8,  the  values  of  A,  B,  and  C,  as  derived  from 

(IV 
the  table,  being  expressed  in  seconds.     To  find  -7-,  we  have 


k{t—T)V\  +  e 
29! 


=  tan^F+itan='^F, 


M'hicli  gives,  by  differentiation, 

k(t-T)         de 


dV 


2qi 


Vl  +  e       cos*'.F' 


and  if  we  introduce  the  expression  for  the  value  of  M  us  ed  a.s  the 
argument  in  finding  V  by  means  of  Table  VI.,  the  result  is 


Hence  we 

dv  _Mc(. 
rfe~750 

by  means 

When  t 
that  the  t 

liie  express 

be  derived 
first  of  the 
regarded  ai 


If  we  tak( 
differentijit 


To  find  th( 
sufficie'it  a 

which  give 

The  equati 


gives 


and  hence 


Snbstitutiii 
dv 


DIFFERENTIAL   FORMULA-, 

Mcos*yV 


133 


Hence  we  have 


dV 

de 


7o(l  +  e)- 


dv     3/ cos* A F        200^  4005    ,,„^.,         600C    ,,^^.„     ,^^, 

-=■---     /    — -prT~vi 7-,   , — ^^7.(1000 -p-,    .,(1000',    (35) 

de     75(1  +  6)     s(l  +  .'0       s(l4-e)^^        ^      8(1 -f  e)'  />     v     / 

by  means  of  which  the  \ahie  of  —  is  readil"  found. 

de 

When  the  eccentricity  differs  so  much  from  that  of  the  jiarabola 
that  the  terms  of  the  last  equation  are  not  sufficiently  convergent, 

liie  expression  for  — ,  which  will  furnish  the  required  accuracy,  may 

be  derived  from  the  equations  (75)i  and  (76)i.  If  we  differentiate  the 
first  of  these  equations  with  respect  to  e,  since  B  may  evidently  be 
regarded  as  constant,  we  get 

dw        ,.  k(t  —  T)  cos^Aiw  ,„„. 

^  (36) 


de 


lO 


l/2qi      i?i/,>^(l  +  9e) 


If  we  take  the  logarithms  of  both  members  of  equation  (76)i,  and 
differentiate,  we  get 


dv   dC        dw 

i?in  y        C       sinu' 


4de 


(l  +  r)(l  +  9e) 


(37) 


To  find  the  differential  coefficient  of  C  wi;h  respect  to  e,  it  will  be 
sufficie.'it  CO  take 

_  1    4  J 


wiiich  gives 

The  equation 

gives 

dA=- 

and  hence  wo  obtain 

dC_ 

G~ 


-^=lC^dA. 

.       5(1 -e)^     ,, 
^^(iT9e)*^''^^*' 

(l4-9e)'         ^  tan  ^i«  COS' J, to 


^;]  +  9tT7  **^"  \wdeAr\   •       dto. 


sm  w 


Substituting  this  value  in  equation  (.'^7),  we  get 


dv 
de 


(l+de) 


sin  V  tan'  ^w  + 


4  sin  V 


C  sin  V    dw 

~8hi7<r  ■  de  ~  (1  4-  eKl  +  9e) ' 


•4  t\    ' 

J.U1: 


THEORETICAL   ASTRONOMY. 


and  substituting,  finally,  the  value  of  -r-,  wo  obtai.i 


dv_  y     k(t—T) 

—  20  • 


C^'sinv 


20  C» 


de 


4  sin-y 


-  smvtanMu; 


(l  +  e)(l+9e)' 

which,  by  raeans  of  (76)j,  reduces  to 
k{t—T)  C'sint) 


^''_    9 


cos':J?<; 


8  tan  4^' 


V2(^      ^v/Jjj(l  +  97)    tan^i,;      (i  _(_  e)  (1  +  9e) 


(38) 


If  we  introduce  the  quantity  M  which  is  used  as  the  argument  in 
finding  w  by  means  of  Table  VI.,  this  equation  becomes 


dv 


9 


3/cos^  Iw 


de      2  (1  -f  9e)    75  tan  ^^o 


C*sinv- 


8  tan  J,v 


(l+e)(l+9ey 


(,;:!9) 


This  equation  remains  unchanged  in  the  case  of  hyperbolic  motion, 
the  value  of  C  being  taken  from  the  column  of  the  table  which  cor- 
responds to  this  case-;  and  it  will  furnish  the  correct  value  of  y-  in 
all  cases  in  which  the  last  term  of  equation  (23)  is  not  conveniently 
applicable.     The  value  of  --  is  then  given  by  the  equation  (32). 

When  the  eccentricity  diifers  very  little  from  unity,  we  may  put 

B  =  \,  and 

tan  ho  =  tan  ^v  i/j^  (i  _^  9^), 

cos''  \w  =  C  cos'  \v. 
Then  we  shall  have 


f-  C  sm  V  = 7--^, —  cos*  iiv. 


The  equation 


gives 


75  tau-^w 


V2q 


Hence  we  derive 


(1  +  A  C)  cos'  >  =  (1  +  ^A)  cos^w, 


l]  =  (1  +  p)  cos*  ^iv  =  Ccos*  iw. 


3/ cos'  Xw 


75  tan  itv 


'  C^sinv- 


k(t  —  T)\/ 


P 


4 


0\ln-  c)- 


NUMERICAL   EXAMPLES. 


135 


If  we  substitute  this  value  in  equation  (39),  and  put  C"  (1  +  e)  =  2, 
wc  get 


de 


9 


kVp 


(.t-T)-: 


8  tan  J,v 


2(1 +  9e)       r'     '"       ^'       (l  +  e){l  +  9e)' 
and  when  e  =-  1,  this  becomes  identical  with  equation  (31). 


(40) 


51.  Examples. — We  will  now  illustrate,  by  numerical  examples, 
the  formulae  for  the  calculation  of  the  variations  of  the  geocentric 
right  ascension  and  declination  arising  from  small  increments  assigned 
to  the  elements.  Let  it  be  required  to  find  for  the  date  18G5  Feb- 
ruary 24.5  mean  time  at  Washington,  the  differential  coolHcients  of 
the  right  ascension  and  declination  nf  the  planet  Eartjnomc  @  with 
respect  to  the  elements  of  its  orbit,  using  tiie  data  and  results  given 
in  Art.  41.     Thus  we  have 

a  =  181°  8'  29".29,      .5  =  —  4°  42'  21".56,     log  J  =  0.2450054, 

logr  =  0.4'28285,  v  =  129°  3'  50".5,  %  =  326°  41'  40".l, 

A  =  '.^96°  39  5".0,       B  =  205°  55'  27".l,  C=  21 2°  32'  17".7, 

log  sin  a  =.  9.999716,         log  sin  6  =^  9.974825,         log  sin  c  =  9.522219, 

log  X  =  0.425066„,  log  y  =  9.51 1920,  log  s  =  8.077315, 

£  =  23°  27'  24".0,  t—T=  420.714018. 

First,  by  means  of  the   equations  (4),  we  compute   the   following 
values : — 


da 


log  cos  <J-r-=  8.054308, 
*  dx 

log  cos  .5^  =  9.7 5491 9„, 


log 


dx 
dl 
'chj 

dz 


Jogl-==  8.668959, 
log --;^  =  6.968348 


n' 


9.753529. 


Tl  ;ni  we  find  the  differential  coefficients  of  the  heliocentric  co-ordi- 
1  ..'S,  with  -espect  to  tt,  Q,,  i,  v,  and  r,  from  the  formula)  (7),  which 
give 


log 


dx 
dr: 


log-^=  <••  <  91991  , 
^  dv 


log  -  / 
dn 


log  ^^  =  0.399496. 

dv  " 


log  J~  -  log  ^^  =  9.950466,., 


log -/'^  =  7. 876553,      log -j|-  =  8.8309 11,    log  ^^'^- =  9.222898„, 


iOg 


dx 
di 


dy 


dz 


=  8.726364,      log  -.^:-  =  9.687577,    log   "%  =  0.142443., 


di 
dy 


di 
dz 


log-^  =  9.996780„,    log -.^  =  9.083635,    log -^— =  7.649030, 


dr 


dr 


dr 


1S6 


THEORETICAL   ASTRONOMY. 


In  computing  the  values  of  -f^>'-r>  and  -v.,  those  of  cos  a,  cosi, 

finil  cos  c  may  gcncmliy  be  obtained  with  sufficient  accuracy  from 
sin  a,  sin  6,  and  sine.  Their  algebraic  signs,  however,  must  be 
strictly  attended  to.  The  quantities  sin  a,  sin  6,  and  sin  c  are  always 
positive;  and  the  algebraic  signs  of  cos  a,  cos  6,  and  cose  are  indicated 
at  once  by  the  equations  (lOl),,  from  which,  also,  their  numerical 
values  may  be  derived.  In  the  case  of  the  example  proposed,  it  will 
be  observed  that  cos  a  and  cos  b  are  negative,  and  that  cos  c  is  positive. 

To  find  the  values  of  cos  d  ^-  and  -j->  we  have,  according  to  equa- 
tion (2), 


da  ,  da     dx    ,  .  da     dii 

4-  cos  5  --        ^ 


C0S5-T-  =  C08-    ,  , 

a::  ax     an 


which  give 


d8 
dn 


dS     dx 

. a — i— 

dx     dn 


dy     dr: 

d8     dy       dS     dz 


(41) 


.da  .  da 

cos  «^—  =  COS  0-7- 

rfrr  dv 


+  1.42345, 


dy     dtc 

d^_dd 
dn        dv 


+ 


dz     dn' 


=  —  0.48900. 


In  the  case  of  Q,  i,  and  r,  we  write  these  quantities  successively  in 
pljice  of  re  in  the  equations  (41),  and  hence  we  derive 


cos  I 


cos^ 


da 

'da 

da 

di 


,    da 

COS  0 r—    = 

dr 


=  —  0.03845, 
^  —  0.27641, 
0.08020, 


dd 

da 

dd 
di 
dS 
dr 


=  —  0.09533, 
=  —  0.78993, 
=  +  0.04873. 


Next,  from  (16),  we  compute  the  following  values  :- 


log  ^  =  0.179155, 
d<p 

log  ^  =  0.171999, 
d^ 


log 
log 


dr 

dM, 

dv 

dK 


9.577453, 


9.911247, 


log 
log 


dr 

d/i 
dv 
dji 


:  2.376581,, 
2.535234. 


dx     dx 


We  may  now  find  -p,  -7^7,  &c.  by  means  of  the  equations  (11), 

and  thence  the  values  of  cos  8  -7-,  -7—,  &c. ;  but  it  is  most  convenient 

a<p    dip 

to  derive  these  values  directly  from  cos  ^-7-,  cos5-i->  -7-,  and  -ry 

''  dr  dv    dr  dv 

in  connection  with  the  numerical  values  last  found,  according  to  the 


NUMERICAL  EXAMPLES. 


137 


equations  wliich  result  from  the  analytical  substitution  of  the  expres- 

-      dx    dy    dz     „       .  .-      /-,\        .^.  •     i  ^r 

sioiis  for  -7-.  -r->  -j-t  okc.,  in  equation  (2),  writing  successively  ^,  M^, 

and  fJt  in  place  of  z.     Thus,  we  have 

.da,  ,  da     dr    ,         .dadv 

cos  0  -  —  =  cos  0  -—  • \-  cos  0  —■ J— 

d^  dr    dtp  dv    d(p 


dS  dS     dr        da     dv 

d<f       dr     dy       dv     dip 


(42) 


and  similarly  for  M^  anrl  'fy  which  give 

cos3-^=+ 1.99400, 
dy 

cos  <5-J^  =  + 1.13004, 


cos<J 


dM, 
do. 
dp. 


+  507.264, 


do 

dip 

dS 

dM, 

J'L 

dfi 


=  —0.65307, 
=  —  0.38023, 
=  —  179.315. 


Therefore,  according  to  (1),  we  shall  have 


cos<5Att  =  + 1.42345a-  — 0.03845Aa  —0.2764Ui     +1.99400a^ 

+  1.13004a  J/„  +  507.264A/i, 
a5  =  —  0.48900^  -  —  0.09533a  S2  —0.78993At    —  0.65307av» 

—0.38023a  J/„— 179.315a//. 

To  prove  the  calculation  of  the  coefficients  •  1  these  equations,  we 
assiiin  to  the  elements  the  increments 


aJ/,  =  +10",  at  =  —  20", 

Ap  =  +  10", 

SO  that  they  become 


Aft  =-10", 
A.a  =  +  0".01, 


Ai  =  +  10", 


Epoch  =  1864  Jan.  1.0  Greenwich  mean  time. 
3/0  =      1°  29'  50".21 
TT  =   44    20  13  .09 1 

ft  =  206    42  30  .13  V  Mean  Equinox  1864.0 
i  =     4    37    0  .51 J 
^0=   11    16    1  .02 
log  a  =  0.3881288 
/x  =  928.56745 

^^^'t!i  these  elements  we  compute  the  geocentric  place  for  1865  Feb- 
ruary 24.5  mean  time  at  Washington ;  and  the  result  is 

tt  =  181°  8'  34".81,        (J  =  —  4°  42'  30".58,        log  J  =  0.2450284, 


'•  I 


14 


138 


THEORETICAL   ASTRONOMY. 


■which  are  referred  to  the  mean  equinox  and  equator  of  1865.0.  The 
ditferenee  between  these  vaUies  of  a  and  (5  and  those  already  given,  as 
derived  from  the  unchanged  elements,  gives 


Aa  =^  +  5".52, 


cos  5  Att  =  +  5".50, 


A(J  =  —  9".02, 


and  the  direct  substitution  of  the  assumed  values  of  at,  aSI,  Ai,  &q. 
in  the  equations  for  cos  3  Aa  and  ao,  gives 


cos  '5  Att  =  +  5".46, 


A<5  =  —  9".29. 


The  agreement  of  these  results  is  sufficiently  close  to  show  that  the 
computation  of  the  differential  coefficients  has  been  correctly  por- 
fornjcd,  the  difference  being  due  chiefly  to  terms  of  the  second  order. 

When  the  differential  coefficients  are  required  for  several  dates,  if 
we  compute  their  values  for  successive  dates  at  equal  intervals,  the 
use  of  differences  will  serve  to  check  the  accuracy  of  the  calculation; 
but,  to  provide  against  the  possibility  of  a  systematic  error,  it  may  l)e 
advisable  to  calculate  at  least  one  place  directly  from  the  changed 
elements.  Throughout  the  calculation  of  the  various  diffi'rential 
coefficients,  gi'cat  care  must  be  taken  in  regard  to  the  algebraic  signs 
involved  in  the  successive  numerical  substitutions.  In  the  example 
given,  we  have  employed  logarithms  of  six  decimal  places;  but  it 
would  have  been  sufficient  if  logarithms  of  five  decimals  had  been 
used;  a..d  such  is  generally  the  case. 

It  will  be  observed  that  the  calculation  of  the  coefficients  of  Ar, 
aSI,  and  Ai  is  independent  of  the  form  of  the  orbit,  depending 
simply  on  the  position  of  the  plane  of  the  orbit  and  on  the  position 
of  the  orbit  in  this  plane.  Hence,  in  the  case  of  i)arabolic  and 
hyperbolic  orbits,  the  only  deviation  from  the  process  already  illus- 
trated is  in  the  computation  of  the  coefficients  of  the  variations  of 
the  elements  which  determine  the  magnitude  and  form  of  the  orbit 
and  the  position  of  the  body  in  its  orbit  at  a  given  epoch.     In  all 

cases,  the  values  of  cos^^-j  cos^-v-?  -r>  and  -r-  are  de<-erMined  as 

av  dr    dv  dr 

already  exemplified.     If  we  introduce  the  elements  T,  q,  and  c,  we 

shall  have 

.da  ^  da     dr    ,         .  da     dv 

cos5^-j,  =  cos^-^--.^y  +  cos^^-.^ 


dS^ 
dT 


dd_ 
dr 


dT'^  dv 


dv^ 
dt' 


(43) 


into  account. 


and  similarly  for  the  differential  coefficients  with  respect  to  q  and  e.   H  the  mass  beins 


nu>ip:rical  examples. 


139 


^,  ,      «     1    ,    .        ,         ,         ^  (^f    (ft'    (^>'  f'''   (ft"        1  (Jf 

The  mode  oi  calculating  the  values  oi  -r™,  -r^,  -,-'  :r'   T'  ^u*^'  t" 

"  ar    (IT   dq    aq    cle  de 

(lepeiids  on  the  natux'c  of  the  orbit. 

In  the  case  of  passhijij  from  one  system  of  parabolic  elements  to 
anotiier  system  of  parabolic  elements,  the  coefficients  of  ac  vanish. 

To  ilhistrate  the  calculation  of  -rph'  "Tm'  &c.  hi  the  case  of  parabolic 

dT  dT  '■ 

motinii,  let  us  resume  the  values  t  —  7^=  75.364  days,  and  log«^ 
=  l).l)()5048G,  from  which  wc  have  found 

logr  ^  0.1961120,  v  =  79°  55'  57".26. 

Then,  by  means  of  the  equations  (22),  we  find 


dr 


log~  =  8.095802„. 
log~=7.976397„, 


dr 

'^dq 
dv 


log  ^^=:.  9.242547, 


log  ^  =  0.064602. 
^dq 


If,  instead  of  dq,  we  introduce  d  log  q,  we  shall  have 


log 


dr 


d  log  q 


9.569812, 


log  ,  f^ -=  0.391867, . 
°  d  log  q  " 


From  these,  by  means  of  (4.3),  we  obtain  the  dilferential  coefficients 
of  a  and  d  with  respect  to  Tand  q  or  logf/.  The  same  values  are 
also  used  when  the  variation  of  the  parabolic  eccentricity  is  taken 

into  account.  But  in  this  case  we  compute  also  j-  from  equation 
(31)  and  ^  from  (33)  or  (34),  which  give,  for  v  =  79°  55'  57".3, 


de 


log|==8.147367„, 


log  ^-==9.726869. 
de 


In  the  case  of  very  eccentric  orbits,  the  values  of  -r^,  -7^,  &c.  are 


found  from 


dT  dT 


dT'~ 

ih_ 

dq'' 


kV 


P 


dT 


V]} 


esmv. 


(44) 


qr"  dq        q       "^      qyp 

dr  r   ,  r'esinv    dv 

dq       q  p       '  dq 


the  mass  being  neglected. 


140 


THEORETICAL  ASTRONOMY. 


To  illustrate  the  application  of  these  formula?,  let  us  resume  the 
values,  <—y-  68.25  days,  c  ^-- 0.9675212,  and  log*/ =^  9.7668134, 
from  wliieh  we  have  found  (Art.  41) 


V  =  102"  20'  52".20, 
Hence  we  derive 

and 

log|^-7.943137„, 

log  ^  =  0.186517., 


log  r  =  0.1614052. 


log  J)  =  0.0607328, 


log 


dT' 
dr 


8.180711  , 


log  3^  =  0.186517, 
^  dq 


If  we  wish  to  obtain  the  diiferential  coefficients  of  v  and  r  with 
respect  to  log  5  instead  of  5,  we  have 


dv 

dlogq 


q     dv 
^ "  dq' 


dr 


q    dr 
d\ogq~\'  dq 


in  which  }.q  is  the  modulus  of  the  system  of  logarithms. 

Then  we  compute  the  value  of  -7-  by  means  of  the  equation  (i 
(35),  (39),  or  (40).     The  correct  value  as  derived  from  (39)  is 


dv 
de 


0.24289. 


i 


The  values  derived  from  (35),  omitting  the  last  term,  from  (40)  and 
from  (30),  are,  respectively,  —  0.24440,  —  0.24291,  and  —  0.23531. 
The  close  agreement  of  the  value  derived  from  (40)  with  the  correct 
value  is  accidental,  and  arises  from  the  particular  value  of  v,  which 
is  here  such  as  to  make  the  assumptions,  according  to  which  equation 

(40)  is  derived  from  (39),  almost  exact. 

dr 
Finally,  the  value  of  y  may  be  found  by  means  of  (32),  which 

gives 

^  =  +  0.70855. 
de 

When,  in  addition  to  the  differential  coefficients  which  depend  on 
the  elements  T,  q,  and  e,  those  which  depend  on  the  position  of  the 
orbit  in  space  have  been  found,  the  expressions  for  the  variation  of 
the  geocentric  right  ascension  and  declination  become 


NUMERICAL   EXAMPLES. 


141 


COS  ')  Att  =  COS'>  --  ATT  4-  COS  0  - —  A  SI  +  COS  ')  -.-  Al  -f  COS  ')"  -  _,  A  T 
dr.  d^  di  dl 


.da 


da 


4-  cos  O  -r-  AO  +  COS  '>  T-  Ae 

da    ^  de 


If  we  introduce  log  q  instead  of  q,  the  terms  containing  q  become 

respectively   cos5-ri — -aIoko    and     ,,        aIosjo.     It  should    be 
'  -^  dlogq        *'  -'  rf  log  7 

observed  that  if  at,  aQ,,  and  a/  are  expressed  in  seconds,  in  order 
that  these  equations  may  be  homogeneous,  the  terms  containing  a 7', 
dq,  and  Ae  must  be  multiplied  by  206264.8;  but  if  at,  aJJ,  and  Ai 
are  expressed  in  parts  of  the  radius  as  unity,  the  resulting  values  of 
cos  0  Att  and  a5  must  be  multiplied  by  206264.8  in  order  to  express 
tlicin  in  i^'econds  of  arc. 

The  most  general  application  of  the  equations  for  cos  d  Aa  and  a5 
in  terms  of  the  variations  of  the  elements  is  for  the  cases  in  which 
the  values  of  cos  8  Aa  and  of  A«5  are  already  known  by  comparison 
of  the  computed  place  of  the  body  with  the  observed  place,  and  in 
which  it  is  I'equired  to  find  the  values  of  A?r,  aSJ,  aj,  etc.,  which, 
being  applied  to  the  elements,  will  make  the  computed  and  the 
observed  places  agree.  When  the  variations  of  all  the  elements  of 
the  orbit  are  taken  into  account,  at  least  six  equations  thus  derived 
are  necessary,  and,  if  more  than  six  equations  are  employed,  they 
must  first  be  reduced  to  six  final  equations,  from  which,  by  elimina- 
tion, the  values  of  the  unknown  quantities  a;:,  aS^,  &c.  may  be 
found.  In  all  such  cases,  the  values  of  Aa  and  a5,  as  derived  from 
the  comparison  of  the  computed  with  the  observed  place,  are  ex- 
pressed in  seconds  of  arc;  and  if  the  elements  involved  are  expressed 
in  seconds  of  are,  the  coefficients  of  the  several  terms  of  the  equations 
must  be  abstract  numbers.  But  if  some  of  the  elements  are  not 
expressed  in  seconds,  as  in  the  case  of  T,  q,  and  e,  the  equations 
formed  must  be  rendered  homogeneous.  For  this  purpose  Ave  nud- 
tiply  the  coefficients  of  the  variations  of  those  elements  which  are 
not  expressed  in  seconds  of  arc  by  206264.8.  Further,  it  is  gene- 
rally inconvenient  to  express  the  variations  aT,  A7,  and  ac  in  parts 
of  the  units  of  T,  q,  and  e,  respectively ;  and,  to  avoid  this  incon- 
venience, we  may  express  these  variations  in  terms  of  certain  parts 
of  the  actual  units.  Thus,  in  the  case  of  T,  we  may  adopt  as  the 
unit  of  A  2'  the  «th  part  of  a  mean  solar  day,  and  the  coefficients 
of  the  terms  of  the  equations  for  cos  d  Att  and  a5  which  involve  aT 


142 


THEORETICAL   ASTRONOMY. 


must  evidently  bo  divided  by  n.  In  the  same  manner,  it  appears 
that  if  we  adopt  as  the  unit  of  A7  the  unit  of  the  »«th  (r '•iniiil 
place  of  its  value  expressed  in  j)arts  of  tin?  unit  of  7,  wc  must  divide 
its  eoetticient  by  10'",  and  similarly  in  the  case  of  Ac,  so  that  the 
equations  beeomc 

cos  O  Ao  =  COS  "  —  At:  4-  COS  o  -—  A  Q  +  cos  o  —^  Al  +  -  cos  «  -,  ,„  A  1 

ih  (IQ,  '  (/t        '   n         clT 


8  .fla  ,        «  ,(/<* 

+  i-.r  COS  rt  ~  Ar/  +  - — ;  cos  o  —-  Ac, 
A5  =  —  A?r  -^ A  O  4 Ai  +  -  .  —  A  r  -^ Afl 


(40) 


in  which  s  =  206264.8.  When  lofj  q  is  introduced  in  place  of  q,  the 
coetfieients  of  A  lojjj  q  arc  multiplied  by  the  same  factor  as  in  the  eas^e 
of  Aq,  the  unit  of  a  \ogq  being  the  unit  of  the  mth  decimal  place 
of  the  logarithms.  The  equations  are  thus  rendered  homogeneous, 
and  also  convenient  for  the  numerical  solution  in  finding  the  values 
of  the  unknown  quantities  att,  aJ2,  a?,  aT,  &c.  When  aT,  a«/,  and 
Ae  have  been  found   by  moans  of  the  equations  thus  formed,  tlie 

AT    Aft 

coirections  to  be  applied  to  the  corresponding  elements  are 


n 


IT 


and 


Ae 
10^ 


In  the  same  manner,  we  may  adopt  as  the  unknown 

quantity,  instead  of  the  actual  variation  of  any  one  of  the  elements 
of  the  orbit,  n  times  that  variation,  in  which  case  its  coefficient  iu 
the  equations  must  be  divided  by  w. 

The  value  of  Aa,  derived  by  taking  the  difference  between  the 
computed  and  the  observed  place,  is  affected  by  the  uncertainty 
necessarily  incident  to  the  determination  of  a  by  observation.  The 
unavoidable  error  of  observation  being  supposed  the  same  in  the  case 
of  a  as  in  the  case  of  d,  when  expressed  in  parts  of  the  same  unit, 
it  is  evident  that  an  error  of  a  given  magnitude  will  produce  a 
greater  apparent  error  in  a  than  in  o,  since  in  the  case  of  a  it  is 
measured  on  a  small  circle,  of  which  the  radius  is  cos  d ;  and  hence, 
in  order  that  the  difference  between  computation  and  observation  in 
a  and  d  may  have  the  same  influence  in  the  determination  of  the 
corrections  to  be  applied  to  the  elements,  we  introduce  cos  0  Aa 
instead  of  Aa.  The  same  principle  is  applied  in  the  case  of  the 
longitude  and  of  all  corresponding  spherical  co-ordinates. 


DIFFEUENTIAL   FORMULAE. 


143 


62.  The  formiilie  nlrondy  given  will  determine  also  tlie  variations 
of  the  geocentric  longitude  and  latitude  corresponding  to  small  in- 
oroiuents  assigned  to  the  elements  of  the  orbit  of  a  heavenly  body. 
Ill  this  case  we  put  e -- 0,  and  compute  the  values  of  A,  11,  siurt, 
and  sin/>»  by  means  of  the  equations  (J>4)i.  We  have  also  ('-  0, 
.*iii  (•  -  sin  /,  and,  in  place  of  a  and  o,  respectively,  we  write  /  and  /5. 
But  when  the  elements  are  referred  to  the  same  fundamental  plane 
as  the  geocentric  phuies  of  the  body,  the  formulie  which  depend  on 
the  position  of  the  plane  of  the  orbit  may  be  put  in  a  form  which  is 
more  convenient  for  numerical  api)lic!ation. 

If  we  ditferentiate  the  etpiations 


we  obtain 


x'  =^r  cos  it  cos  Q,  —  r  sin  u  sin  Q,  cos  i, 
•if  z=r  cos  «  sin  ft  +  »•  sin  u  cos  ft  cos  i, 
/  =  r  sin  tt  sin  i, 


X 


dx'^-dr 
r 


rfsiu  u  COS  ft  -f"  coau  sin  ft  cos  O^^^ 
—  r(cos  n  sin  ft  +  sin  u  cos  ft  cos  i)  dSl  -\-r  sin  u  sin  ft  sin  i  di, 

v' 

dy'  =  —  dr  —  r  (sin  ?t  sin  ft  —  cos  u  cos  ft  cos  0  du 

■\-  j*(co3  u  cos  ft  —  sin  n  sin  ft  cos  i)  dQ,  —  r  sin  «  cos  ft  sin  i  di,  (46) 

z' 
dz'  =  -  dr  -{-  r  cos  u  sin  i  du  +  r  sin  u  cos  i  di, 

in  which  x',  y',  z'  are  the  heliocentric  co-ordinates  of  the  body  in 
reference  to  the  ecliptic,  the  positive  axis  of  x  being  directed  to  the 
vernal  equinox.  Let  us  now  suppose  the  place  of  the  body  to  be 
referred  to  a  system  of  co-ordinates  in  which  the  ecliptic  remains  as 
the  plane  of  xxj,  but  in  which  the  positive  axi3  of  x  is  directed  to  the 
point  whose  longitude  is  ft  ;  then  we  shall  have 

dx  =  d^  cos  ft  +  d^  sin  ft , 
rfy  =  —  t/^'  sin  ft  +  d^  cos  ft , 

dz  =  (?3', 

and  the  preceding  equations  give 

dx^=-dr  —  r  sinu  du  —  r  ainu  cosi  dft, 
r 

dy  =  ^  dr  -{-  r  cos u  coai  du  -{-  r  coau  d^l  —  r  "in u  sin i  di,    (47) 


life 


dz  =  -dr  -\-r  coau  sin idu  -\-r  sin u  cos i  di. 
r 


144 


TIIEORETK'AT.    ASTItONOMY. 


This  traiisfornintion,  it  will  he  «)1)s(M'V('i1,  is  ('f|uiv{ilcnt  to  (liminisliini^ 
the  lonjiitiidort  in  the  oqimtious  (4G)  by  the  angle  Q  tiirough  which 
tlie  axis  of  x  has  boon  moved. 

Let  X„  Y,y  %,  denote  the  heliocentric  co-ordinates  of  the  earth 
referred  to  the  sanuj  system  of  co-ordinates,  and  we  liavo 

X -\-  X,  =  J  cos li  cos  (i  —  SI), 
y+  r,=  Jcos/9  8in(A— Q), 
2  -f  Z,  =  J  sin  /9, 

in  whicli  ?.  is  the  geocentric  longitude  and  /9  the  geocentric  latitude. 
In  ditferentiating  these  equations  so  as  to  find  the  relation  between 
the  variations  of  the  heliocentric  co-ordinates  and  the  geocentric  lon- 
gitude and  latitude,  we  must  regard  ft  as  constant,  since  it  indicates 
here  the  position  of  the  axis  of  x  in  reference  to  the  vernal  equinox, 
and  this  position  is  supposed  to  be  fixed.     Therefore,  we  shall  have 

r?.f  =  cos/5  cosU  —  Sl)<l-^  —  "J  sin /9  cos  (/^ — SDdi^  —  ^  cos /?  sin  (A — Sl)(U, 
t/?/=  cos (S  sin  (f.  —  ft  )  rf-J  —  -J  sin /9  sin  (A — ft  )  rf/J  -(-  J  co8/3  cos  (A  — ft ) tU, 
dz  =8in  ,3  dJ  +  J  cos/J  d,3, 


from  which,  by  elimination,  we  find 

sin  (A 


cos  /J  dl  ■■ 


^dx-^-'-^^ 


ft) 


d>3  =  . 


sin,?  cos  (A  — ft) 


dx- 


dy, 


sin/JsinfA— ft)       ,  cos/5  , 
-. dy+     .    dz. 


These  equations  give 


.dX 
cos  /9  -,—  = 
ax 


sin  (A  —  ft) 


COS/? 


dk       cos  (A— ft) 


dy 

cos  /?   ,  =  0, 
dz 


dlS 
dx 
dl 
dy 

P. 
dz 


sin/?cos(A  —  ft) 

J  ' 

sin/?  sin  (A  —  ft) 

C08/5 


>  (48) 


If  we  introduce  the  distance  to  between  the  ascending  node  and  the 
place  of  the  perihelion  as  one  of  the  elements  of  the  orbit,  we  have 

dii.  =  dv  -\-  d(a, 

and  the  equations  (47)  give 


dx       x 

-r =  _  =  cos  U, 

dr       r 
dx        dx 


dy       y        . 
— , — = -  =  sin  M  cos  I, 
dr       r 

dy      dy 


dz 
dr 
dz        dz 


:_  =  sm«  sm  i; 
r 


,  -=  ,    ==  —  rsmn,  -  ,"-=  /  =-r cos w cost,  -j— =-y-=rcos«sini; 
dv       duj  dv       dto  dv       aw 


DIFFERENTIAL   FORMULAE. 


(Ix  .  .      (hi 

rfft  'da 


j'-'  =0 


dy  ... 

■  ,. -  =  —  r  Sinn  sin  », 
di 


da 
dz 

di 


=--:0; 


145 

(49) 


-  r  sm  n  cos  i. 


If  wc  introduce  r,  the  longitude  of  the  perihelion,  wo  have 

du  =  dv  -\-  dir  —  da, 

and  hcnro  the  expressions  for  the  partial  dift'orontial  eoofficionts  of 
the  heliocentric  eo-ortUnatcs  with  respec^t  to  z  and  a  become 


dx 
dx 


r  sin  u, 


3^L 

di: 
dy 


r  cos  u  cos  t, 


2r  sin  u  sin'  ^i,        '    -—  2r  cos  u  sin'  J «, 


dn 

dz 

da 


— , —  =  r  cos  M  sin  i ; 
•^  (50) 

=  —  r  cos  K  sin  i. 


When  the  direct  inclination  exceeds  90°  and  the  motion  is  rej];arded 
a.s  being  retrograde,  we  find,  by  making  the  necessary  distinctions  in 
regard  to  the  algebrai(!  signs  in  the  general  equations, 


dx 


di 


r  =  0, 


di 


r  sin  tt  sin  i. 


dz 
di 


—  r  sin  M  cost;      (51) 


and  the  expressions  for  -y^,  -j-,  -r^T'  'i~'  ^^'  ^^^  derived  directly 

from  (49)  by  writing  180°  —  i  in  place  of  i.     If  we  introduce  the 
longitude  of  the  perihelion,  we  have,  in  this  case, 


and  hence 

dx 
—  -  =  r  sin  u, 

dx 


du  ===  rfv  —  di:  +  da> 

dn 
dy 


,— -  =  r  cos  «  cos  I,       — ,—  =  —  r  cos  v,  sm  i ; 
^'^  ^'^  (52) 

r  cos  w  sin  i. 


dz 
dn 
dz 


But^  to  prevent  confusion  and  the  necessity  of  using  so  many  for- 
miihe,  it  is  best  to  regard  i  as  admitting  any  value  from  0°  to  180°, 
and  to  transform  the  elements  which  are  given  with  the  distinction 
of  retrograde  motion  into  those  of  the  general  ease  by  taking 
180°  —  i  iii«itead  of  i,  and  2a  —  7t  instead  of  -,  the  other  elements 
remaining  the  same  in  both  cases. 

53,  The  equations  already  derived  enable  us  to  form  those  for  the 

differential  coefficients  of  X  and  j3  with  respect  to  r,  v.  a ,  i,  and  w  or 

~,  ])y  writing  successively  k  and  /9  in  place  of  d,  and  a,  i,  &c.  in 

10 


14G 


THEORETICAL    ASTRONOMY. 


pl.'U'O  of  TT  ill  eqiiiation  (2).  The  expressions  for  the  differential  coefii- 
t'ients  of  r  and  c,  with  respect  to  the  elements  which  determine  the 
form  of  the  orl)it  and  the  position  o^  the  body  in  its  orbit,  being 
independent  of  the  position  of  the  plane  of  the  orbit,  arc  the  same  as 
those  already  given;  and  hence,  according  to  (42)  and  (43).  we  may 
derive  the  value ■>  of  the  partial  ditterential  coeiHcients  of  I  and  ^5 
with  respect  to  ihese  elements.  The  numerical  application,  however, 
is  facilitated  by  the  introduction  of  certain  auxiliary  quantities. 
Thus,  if  wc  substitute  the  values  given  by  (48)  and  (49)  in  the 
equations 

dv 


.dX  ^  dk     dx    ,  .dk 

cos  p  -J-  =:  cos  p  -7-  •  -,  -  4-  cos  p-y 

dv  dx     dv  dy 


and  put 


dv 


d,3 
dx 


dx        d,3 
dv        dy 


di       d,3 
dv  ^  rfz 


rf2 

dv' 


COS  i  cos  {X  —  Q)  =^  Ag  sin  A, 


sin  (A  —  SI)  — 


cos. 


H^a  i 


■■  n  sm 


—  sin  (A  —  Q)  cos i  --  n  cos N, 
in  which  Aq  and  n  are  always  positive,  they  become 


(53) 


dX 


dX 


cos  /?—-=:  cos  /?  -p 


dv 
dv 


d3 


■  An  sin  (A  -I  -  u), 


1 

■  —  (sin  /3  cos  (A  —  £1)  siu  n  -\-  n  cos  u  sin  (iV  -}-  ,3)  ). 


Let  us  also  put 
and  wc  have 


n  i^'m  (N -\-  ft) ---^^  Ba  !iin  B, 
sin  fi  cos  (A  —  J^  )  -—  B,.  cos  B, 


cos/3 


dX 
dv 
dfi 


COS  ,3  --  ^.--  -  Ao  sm  (^1  +  u), 


dv-  1=3^-^^-^--)- 


(54) 


The  expressions  foK  co?  ft-j-  and  -,'-  give,  by  means  of  the  sppie 
auxiliary  (piantities, 


.dk 

cos  p  -,- 

dr 


---  cos  (xl  -f  «), 


-j-  = V  cos  {B  4-  ^l). 

dr  J 


m 


In  the  same  manner,  if  we  put 


DIFFERENTIAL  FORMULJS. 


147 


we  obtain 


cos  (X  —  SI)  =  Co  sin  C, 
cos  i  sin  (A  —  ^)  =^  C\  cos  C; 

cos  i  ^=  Z>„  sin  J), 
sin  (A  —  J^)  sin  i=^  D^  cos  X); 

cos  /3  -^  _   =  ^  Co  sin  (  C  +  «), 

^ '  -  =  —  ^  J„  sin  ,5  cos  {A  +  it) ; 

dX  r    .       . 

cos  /?  —J—  =  —  -T  sin  i  sin  u  cos  (A  —  ^  ), 

— — -  =  -  n  sin  «  sin  (D  +  /?). 
di        r     ° 


If  we  substitute  the  expressions  (55)  and  (56)  in  the  equations 

„  di.  .  dX     dr    ,         .  dk     dv 

cos  /?  -.—  -—  cos  li  -y~  •  -T — \-  cos  fl  -, ,— , 

d(p  dr    d<p  dv    d<p 


(57) 


(58) 


d,3_ 
dtp 


and  put 


d'i     dr  _.d,3_    dv 
dr    dip        dv    dtp' 


—  -   -  =f  sin F  =a  cos ^  cos v, 
f  ,  -  =/  cos  jP  =  I  — " — \-  tan  «>  cos  v  1 


(59) 


/•  sni  V, 


wc  get 


cos  ii~=.l-  Ag  sin  ( J.  +  JF  +  ii), 

lu  a  similar  manner,  if  wc  put 

dr  .     „  . 

— 1  nf  =  <7  sm  (r  =  —  a  tan  «>  sm  v, 

dAL      '' 


(60) 


dv 


„       a'  cos  «> 


=  /i  sin  ir.-=  —  I  a  tan  ?>  sin  v(e  —  T)  —  ,J  206264.8  j, 


dr 
~    dji 

dv  ,  rr         «*  COS  ^    .,  ™. 

,. ..       —  fi  eos  JET  = (t  —  T), 

dft  r 


(61) 


148 

we  obtain 


THEORETICAL  ASTRONOMY. 


COS  /?  j^jT  =^  I  Ao  sin  (^  +  G  +  iO, 
^  =  ^-B„8in(5+G  +  «); 

cos/3  -^—  =  -J  ^0  sin  (^  +  ir+  i{)i 
-^  =  ^-J5oSin(£+£r+u). 


(62) 


The  quadrants  in  which  the  auxiliary  angles  must  be  taken  are 
determined  by  the  condition  that  Ag,  Bq,  Cj,,/,  g,  and  h  are  always 
positive. 

54.  If  the  elements  T,  q,  and  e  are  introduced  in  place  of  ilf,,,  //, 
and  (f,  we  must  put 


g  sin  C  =  — 

h  sin  1^=  — 

and  the  equations  become 

dk 


de' 
dr 
df' 
dr 


ff  cos  G^^r 


h  cos  jff  =  r 


dv 

dv^ 
dq' 


(63) 


d,3 
de 


cos  /9  ^  =  ^  Jo  sin  (A-{-  F-\-  iC), 


B^^m^B-\-F-\-xC)', 


(64) 


cos  l^^i=~i  A  sin  ( J  +  G  +  it). 
f^,=  ^B,,sm(B+G  +  u); 

cos  /9  -7-  =:  —  ^0  sin  (J.  +  iT^-  w), 

-^=.-^J5oSin(jB  +  ir+«). 

In  the  numerical  application  of  these  formulre,  the  values  of  the 
second  members  of  the  equations  (63)  are  found  as  already  exem- 
plified fur  the  cases  of  parabolic  orbits  and  of  elliptic  and  hyperbolic 
orbits  in  which  the  eccentricity  differs  but  little  from  unity.  In  the 
same  manner,  the  ditfcrential  coefficients  of  X  and  j3  with  respect  to 
any  other  elements  which  determine  the  form  of  the  orbit  may  be 
computed. 


NUMERICAL   EXAMPLES. 


149 


In  the  case  of  a  parabolic  orbit,  if  the  parabolic  eccentricity  is 
supposetl  to  be  invariable,  the  terms  involving  e  vanish.  Further, 
in  the  case  of  parabolic  elements,  we  have 


which  give 


fj  COS  G  =  r^^, 


^sinv 


r  tan  Iv 


dv 
dl" 


tan  G  =  —  tan  ^v. 
Hence  there   results   G  =  180°  —  ^v,  and  </ 


k\--,  which  is  the 

expression  for  the  linear  velocity  of  a  comet  moving  in  a  parabola. 

Therefore, 

.dX            k\/2    .     .    ,  .    .  ,  . 

cos p ~,^  = r~7-" ^0  sin  (-4  4-  tt  —  \ V), 

(65) 


dT 
dp 


V  r 

kV2 


dT 


= :,—.--  ^0  sin  (B-\-  u~  \v). 


^Vr 


For  the  case  in  which  the  motion  is  considered  as  being  retrograde, 
180°  —  i  must  be  used  instead  of  i  in  computing  the  values  of  A^, 
A,  n,  N,  Cfj,  and  C,  and  the  equations  (55),  (56\  and  the  first  two 
of  (58),  remain  unchanged.  But,  for  the  differential  coefficients  with 
respect  to  /,  the  values  of  />„  and  J)  n\ust  '■  found  from  the  last  two 
of  equations  (57),  using  the  given  value  liroctly;  and  then  we 

sliall  have 


r.  dX       r   .    .  .  ,.       ^  ^ 

cos  /?  -y.  =^  -7  sm  t  sm  u  cos  (/  —  JJ ), 

- ,  r  == T  Da  sm  It  sm  {D  +  ,3). 

d%  J     " 


(66j 


55.  Examples. — The  equations  thus  derived  for  the  diflercniial 
coefficients  of  X  and  /5  with  respect  to  the  elements  of  the  orbit, 
rpferred  to  the  ecliptic  as  the  fundamental  plane,  are  applicable  when 
any  other  plane  is  taken  as  the  fundamental  plane,  if  we  consider  ) 
and  /9  as  having  the  sam<'  signification  in  reference  to  the  new  plane 
that  they  have  in  reference  to  the  ecliptic,  the  longitudes,  however, 
being  measured  from  the  place  of  the  descending  node  of  this  plane 
on  the  ecliptic.  To  illustrate  their  numerical  application,  let  it  be 
miuire<l  to  find  the  differential  coefficients  of  the  geocentric  right 
ascension  and  declination  of  Eunjnomc  ©  with  respect  to  the  ele- 
ments of  its  orbit  referred  to  the  equator,  for  the  date  1865  February 
24.5  mean  time  at  Washington,  using  the  data  given  in  Art.  41. 


150 


THEORETICAL  ASTRONOMT- 


In  the  first  place,  the  elements  which  are  referred  to  the  ecliptic 
must  be  referred  to  the  equator  as  the  fundamental  plane ;  and,  by 
means  of  the  equations  (109)i,  we  obtain 


SI' 
and 


:  353°  45'  35".87,        i'  =  19°  26'  25".76,        w,  =  212°  82'  17".71, 


a,'  =  o,^w^  =  50°  10'  7".29, 


which  are  the  elements  which  determine  the  position  of  the  orbit  in 
space  when  the  equator  is  taken  as  the  fundamental  plane.  These 
elements  are  referred  to  the  mean  equinox  and  equator  of  1865.0. 
Writing  a  and  8  in  place  of  A  and  ,9,  and  Q',  i',  w'  in  place  of  ft,  /, 
and  10,  respectively,  we  have 


Aq  sin  J.  =  cos  (a  —  ft')  cos  i', 
n  sin  N  =  sin  i', 
BoSmB  =  nsm{N+d), 
C[  sin  C=  cos  (o  —  ft'), 
DgSinD--=  cos  /', 

/  sin  F  =^  a  cos  f  cos  v, 

cos^ 
g  sin  (?  =  —  «  tan  f  sin  v, 


A^  cos  J.  =  sin  (o  —  ft') ; 
n  cos  iV=:  —  cos  i'  sin  (o  —  ft ')  ; 
J?j  cos  B  =  sin  ^  cos  (a  —  ft ') ; 
Co  cos  C'=  sin  (a  —  ft')  cos  i' ; 
Do  cos  Z)  =  sin  i'  sin  (a  —  ft ') ; 


/  cos  F- 1 +  tan  cp  cos  v  | 

•'  \  cos  ^  ^         ^  I 


r  sm  v; 


cr  cos  <p 


r 


g  cos  G  ■■ 

h  sin  H=  —  I  a  tan  ^  sin  v  {t  —  T) 


2r 


206264 


J4.8), 


h  cos  if = 


.^-^^lit-T). 


The  values  of  -4^,  ?i,  jB„,  Q,  /)„,  /,  (/,  and  h  must  always  be  positive, 
thus  determining  the  quaih'unts  in  which  the  angles  A,  B,  <&c.  must 
be  taken ;  and  these  equations  give 


log  ^0=9.97497, 
log^o^  9.52100, 
log  C„  -=  9.99961, 
log  Do  =  9.97497, 
log/  =0.62946, 
log  g  =  0.34593, 
log  h  =  2.07759, 


A  =  262°  10'  40", 

B=   75  48  35  , 

C  =  263  2    6, 

/)  =   92  35  47  , 

F  =  339  14    0  , 

0  =  350  11  16  , 

H=   14  30  48  , 


u' 


:v-t-w'=179°  13' 58". 


NUMERICAL   EXAMPLES. 


161 


Substituting  these  valucri  in  the  equations  (55),  (58),  (60),  and  (62), 
and  writing  a  and  d  instead  of  X  and  ^,  and  u'  in  phiee  of  u,  we  find 


cos^ 
cos  3 
cos^ 
cos  5 
cos  5 
cos  (J 


and  hence 


da 

da 

da 
^dV 
da 
dtp 
da 

rfJ/„ 

do^ 
dji 


=  +  1.4235, 
=  +  1.5098, 
=  +  0.0067, 
=  +  1.9940, 
==  +  1.1300, 
=  +  507.25, 


da 

dio' 

dSi 

_^ 
di' 

d3 


d<p 

dS 

dM, 

_dd 


—  0.4890, 
-,  =^  +  0.0176, 

+  0.0193, 

—  0.6530, 


=  —  0.3802, 


— 179.34 ; 


cos  (3  Ao  =  +  1.4235  Mo'  +  1.5098  aJJ'  +  0.0067  Ai'  +  1.9940  Av> 
+  1.1300  Ail/„  +.  507.25  A//, 
A<J  ==  —  0.4890  Aw'  +  0.0176  ^Q,'  +  0.01t3  At'  —  0.6530  a^ 
—  0.3802  aJ/„  —  179.34  A//. 
If  we  put 


Aw'  =:  —  6".64, 

As?    =:.  +  10", 


ASi'=-14".12, 
aJ/„  .-  +  10", 


we  get 


cos  5  Aa  =  +  5".47, 


A'J 


Ai'  =  —  8".86, 

A/t=+0".01, 

9".29 ; 


and  the  values  calculated  directly  from  the  elements  corresponding  to 
the  increments  thus  assigned,  are 


cos  5  Aa  =:  +  5".50, 


a5  =  —  9".02. 


The  agreement  of  these  results  is  s'j  'fioiently  close  to  prove  the  cal- 
culation of  the  coefficients  in  the  equatio..  i  for  cos  d  Aa  and  Ao. 

When  the  values  of  aw',  Aft',  and  Ai'  are  small,  the  correspond- 
ing values  of  aw.  Aft,  and  aj  may  be  detemiined  by  means  of 
(litferential  formula?.  From  the  spherical  triangle  formed  by  the 
intersection  of  the  planes  of  the  orbit,  ecliptic,  and  equator  with  the 
celestial  vault,  we  have 


cos  i  r=  cos i'  cos  e  +  sin  i'  sin  e  cos  ft', 
sin  i  cos  ft  =  —  cos  /'  sin  e  +  sin  i'  cos  e  cos  ft', 
sin  i  sin  ft  =  sin  i'  sin  ft', 
sin  i  sin  w^  =  sin    \ '  sin  e, 
sin  i  cos  w,  =^  cos  s  sin  i'  —  sin  e  cos  i'  cos  ft', 


(.67) 


152 


THEORETICAL  ASTRONOMY. 


from  which  tlic  vahics  of  Sly  i,  and  w„  may  be  found  from  those  of 
JJ '  and  i'.  If  wc  differentiate  the  first  of  these  equations,  regarding 
e  as  constant,  and  reduce  by  means  of  the  other  given  relations,  we 

get 

dt  =  cos m^di' -{- sintu^ sin i'd Si'-  (68) 

Interchanging  i  and  180°  —  i',  and  also  Q  and  SI',  we  obtain 

di'  =  cos  Wfl  di  —  sin  w^  sin  id  SI. 
Eliminating  di  from  these  equations,  and  introducing  the  value 


the  result  is 


sin  i' sin  SI 

siui       siu^'' 

,_        sin  SI  ir^,      sihw    ,., 

dSl  =  -.—}r,  cos lUod SI .     "-  di', 

s\nSl  smi 


(69) 


If  we  differentiate  the  expression  for  cos  Wq  derived  from  the  same 
spherical  triangle,  and  reduce,  we  find 

dujg  =  cos  t  dSl  —  cos  i'  dSl'' 

Substituting  for  dSl  its  value  given  by  the  preceding  equation,  and 
reducing  by  means  of 


we  get 


sin  SI'  cos  i'  =^  sin  SI  cos  ">„  cos  i  —  cos  SI  sin  to^, 


,         smw  ,    ,      smwj  , 

dm  =  -r    .-°  cos  SI  dSl '. — .  cos  I  di. 

"      sm  Si  sm  t 


(70) 


The  equations  (68),  (69),  and  (70)  give  the  partial  differential  co- 
efficients of  SI  J  h  and  Wq  with  respect  to  SI '  and  i',  and  if  wc  sup- 
pose the  variations  of  the  elements,  expressed  in  parts  of  the  radius 
as  unity,  to  be  so  small  that  their  squares  may  be  neglected,  we  shall 
have 


8uiw„        ^     ^,      smw         .    ., 

Aw.  =  -,-  -^»  cos  ft  A  J2 .    ".-  cos  %  hi', 

^       i^in  SI  ^ ,       sin  «>.     ,, 

sm  SI  sm  % 

^i  =  sin  Wj  sin  i'  A  ft '  -(-  cos  w^  Ai', 
Aw  =:=  Aw'  —  Aw.. 


(71) 


If  we  apply  these  formula}  to  the  case  of  Eurynome,  the  I'esult  is 

Aw„  =  — 4.420Aft' +  6.665  Ai', 

Aft  ==  —  3.488a  ft'  +  6.686aj:', 

Ai  =  —  0.179a  ft'  —  0.843Ai'; 


DIFFERENTIAL  FORMULA.  153 

and  if  we  assign  the  values 

Aft'  =  -  14".12,  Ai'  -^  —  8".86,  aw'  =  -  6".64, 

we  get 
Aw„  =  +  3".36,       A  ft  =  —  10".0,       Ai  =  +  10".0,       Aw  ^  —  10".0, 

and,  hence,  the  elements  which  determine  the  position  of  the  orbit  in 
reibrence  to  the  ecliptic. 

The  elements  to',  ft',  and  i'  wvAy  also  be  changed  into  those  for 
which  the  ecliptic  is  the  fundamental  plane,  by  means  of  e(iuations 
wliich  may  be  derived  from  (109),  by  interchanging  ft  and  ft'  and 
180°  — i' and  i. 

5G.  If  we  refer  the  geocentric  places  of  vhe  body  to  a  plane  whose 
inclination  to  the  plane  of  the  ecliptic  is  i,  and  tlie  longitude  of  whose 
ascending  node  on  the  ecliptic  is  ft, — which  is  equivalent  to  taking 
the  plane  of  the  oi*bit  corresponding  to  the  unchanged  elements  as 
the  fundamental  plane, — the  equations  are  4ill  further  simplified. 
Let  x',  y',  z'  be  the  heliocentric  co-ordinates  of  the  body  referred  to 
a  system  of  co-ordinates  for  which  the  plane  of  ihe  unchanged  orbit 
is  the  plane  of  xy,  the  positive  axis  of  x  being  directed  to  the  as- 
cending node  of  this  plane  on  the  ecliptic;  and  let  x,  y,  z  be  the 
heliocentric  co-ordinates  referred  to  a  system  in  which  the  plane  of 
xy  is  the  plane  of  the  ecliptic,  the  positive  axis  of  x  being  directed 
to  the  point  whose  longitude  is  ft .     Then  we  shall  have 

dx'  ==  dx, 

dtf  =  dy  cos  i  -\-  dz  sin  ", 

dz'  =  —  dy  sin  i  -(-  dz  cos  i. 

Substituting  for  dx,  dy,  and  fh  their  values  given  by  the  equations 
(47),  we  get 


dx'  =:  —  di'  —  r  sin  u  du 
r 


r  sinu  cosi  c?ft, 


.?/ 


dy'  = '—  dr  -f-  *'  cos  ii  du  -f  r  cos  ti  cos  i  Jft , 


dz'  =  -  dr 
r 


r  cos  M  sin  i  c?ft  -\-  r  sin  u  di. 


It  will  be  observed  that  we  have,  so  long  as  the  elements  remain 
unchanged, 


r  cos  u, 


i/  =  r  aiii  u, 


z'  =  0, 


154 


THEORETICAL  ASTRONOMY. 


and  henco,  omitting  the  accents,  so  tlmt  x,  y,  z  w\\\  refer  to  the  plane 
of  the  unchanged  orbit  as  the  plane  of  xi/,  the  preceding  equatioiia 
give 

dx  =  cos  it  dr  —  r  sin  u  du  —  r  sin  ri  cos  i  dSl, 
dij  =  sin  It  dr  -\-  r  cos  u  du  -f-  f  cos  a  cos  i  dSl, 
dz  =  —  r  cos  u  sin  i  dSl  +  r  sin  w  di. 

The  value  of  w  is  subject  to  two  distinct  changes,  the  one  arising 
from  the  variation  of  the  position  of  the  orbit  in  its  own  plane,  anil 
the  other,  from  the  variation  of  the  position  of  the  plane  of  the  orbit. 
Let  us  take  a  fixed  line  in  the  plane  of  the  orbit  and  directed  from 
the  centre  of  the  sun  to  a  point  the  angular  distance  of  which,  back 
from  the  place  of  the  ascending  node  on  the  cclij)tic,  we  shall  desig- 
nate by  <t;  and  let  the  angle  between  tiiis  fixed  line  and  the  senii- 
transvcrse  axis  be  designated  by  ^.     Then  we  have 

;|.  =  W  -I-  ff. 

The  fixed  line  thus  taken  is  supposed  to  be  so  situated  that,  so  long 
as  the  position  of  the  plane  of  the  orbit  remains  unchanged,  we  have 

But  if  the  elements  Avhich  fix  the  position  of  the  plane  of  the  orbit 
are  supposed  to  vary,  we  have  the  relations 


da  =cosi  dSl, 

dm  =  dj(  —  cos  i  dSl , 

dx  :=dx  -\-  0-  —  COS  t)  dQ  =  t?/  +  2  sin' U  d^. 

Now,  since  u  =^  v  -{-  w,  we  have 

du  =  dv  -{-  dx  —  dff  :^  dv  -{■  dx  —  cos i  dQ. 


(72) 


and 


Substituting  this  value  of  du  in  the  equations  for  dx,  dy,  dz,  they 

reduce  to 

dx  =  cos  11  dr  —  r  sin  u  dv  —  r  sin  u  dx, 

dy  :=  sin  u  dr  -\-r  cos  it,  dv  +  ''  cos  u  dx,  (73) 

dz  =  —  r  cos  u  sin  i  dQ  +  *'  sin  u  di. 

The  inclination  is  here  supposed  to  be  susceptible  of  any  value  from 
0°  to  180°,  and  if  the  elements  are  given  with  the  distinction  of 
retrograde  motion' we  must  use  180°  —  i  instead  of  i. 

Let  us  now  denote  by  d  the  geocentric  longitude  of  the  body  mea- 
sured in  the  plane  of  the  unchanged  orbit  (which  is  here  taken  as  the 


DIFFERENTIAL   FORMULA. 


165 


fundamental  plane)  from  the  ascending  node  of  this  plane  on  the 
ecli[)tie,  and  let  the  geoeentric  latitude  in  rcl'erence  to  the  same  plane 
be  denoted  by  tj.     Then  we  shall  have 

X  -\-  X=^  ^  cos y}  cos  0, 
y  -\-  F=-  J  cos>j  siu^', 
z  -\-  Z  -^  J  sin  rj, 

in  which  X,  Y,  Zare  the  geocentric  co-ordinates  of  the  sun  referred 
to  the  same  system  of  co-ordinates  as  x,  y,  and  z.  These  equations 
give,  by  ditt'orentiation, 

dx  =  cos  Tj  cosO  (iJ  —  J  sin  ij  cos  0  drj  —  J  cos  rj  sin  0  dO, 
dy  ==  cos  Tj  sin  0  dJ  —  J  sin  ij  sin  0  drj  -{-  J  cos  )j  cos  0  dO, 
dz  =s\nr)  dJ  -}-  ii  cos  rj  drj ; 


and  hence  we  obtain 
cos  Tj  do  z=  — 

drj  =  — 


sin  <?  ,     ,   cos  0  , 
-j'dx-\--^-dy. 

sinijcos^?  ,         siuij  sinO  ,     ,   cosw  , 

—A — ^^- ^i — "^y+'A  d'- 


These  give 


do 


sin  0 


'''''dx  =  — r' 

drj  sin  5j  cos  0 

~dx  ~  J         ' 

and  from  (73)  we  get 


COS)? 

dv 


do 
dy 


_  cos  0 
s\\\yj  sinO 


cos  rj,-  =  0; 
us 


d^ 
dz 


COS)? 


(74) 


dx^ 
dr 

dx 
dv 


■  COS  u, 
dx 


dy 

—f—  =  sm  u, 
dr 


0; 


di        "' 


r  sni  11, 
dy 


dy_ 
dv 


dSl 


-0, 


dy 


'r  =  0, 


dy 

dz 
dl 
dz 


:  r  cos  u, 


dz 
dr 

dz   dz » 

dv   ~'dx~    ' 

(75) 


-j-^-  =^  —  r  COS  u  sm  j; 


=  r  sin  u. 


di  '  di 

Substituting  the  values  thus  found,  in  the  equations 


do  do     dx    ,  do     dy 

cos  rj    ,  =  cos  ij  - , J ^  cos  )7    ,     •    ,   , 

^..  j„     J..  dy     dv 


dv 


dx    dv 


drj rf)j     dx        dyj     dy        drj     dz 

•  dv       dx    dv       dy     dv        dz     dv ' 


156 

wc  get 


THEORETICAL  ASTRONOMY. 


do  do       r         .^         . 

COS  )j    ,  =  cos  w  -y    =^  -J  cos  Kp  —  «), 
dx       J 


dri 


(76) 


,    —    ,    —  —  -r  sin  ij  sin  (0  —  tt). 
In  a  similar  manner,  we  derive 


do  1    .    ,.        X 

cos  ij  — ,  -  =  —  —  sm  {0  —  u), 
dr  J 

do 

CO8,^^^-=0. 

do 


-  ,  -  =  —  -  sm  19  cos  {0  —  u), 
dr  J 

dij  r  .    . 

- , ^  =  —  -J cos ij  sin  t  cos u,   ill) 
dQ  A        ' 

drj  r 

—,.-  =  4-  —  cos «  sm u. 
di  J 


If  we  introduce  the  elements  <p,  M^,  and  /u,  which  determine  r  and  r, 

we  have,  from 

do  do     dr    ,  dO     dv 

cos  IJ  -p-  =  cos  5J  -y-  •    7 \-  COS  17  -f-  •  -J—, 

dtp  dr    d<p  dv    dtp 

drj  dr,     dr        drj     dv 

dtp       dr    dip       dv    d^' 

if  we  introduce  also  the  auxiliary  quantities /and  F,  as  determined 
by  means  of  the  equations  (59), 

cos ri ^  =  I, cos (O  —  ii  —  F),     ^'^'-  =  — l^amrj  sin (0  —  u  —  F).    (78) 
11^       J  a<p  J 

Finally,  using  the  auxiliaries  j,  h,  G,  and  H,  according  to  the  equa- 
tions (61),  we  get 


COSJ? 


do 
dM, 


cos  ip  —  «  —  G), 


cos 


J 
Ij    -=  —  coa{0  —  u  —  H), 


dr) 

dji; 

dfi 


—  ^ sin ij  sin  (^  —  u—tG), 

h  .  ^''^^ 

—  -j^  sin  ij  sin  (6>  —  u  —  II). 


If  we  expi'ess  r  and  v  in  terms  of  the  elements  T,  </,  and  e,  the 
values  of  the  auxiliaries  /,  g,  h,  F,  &c.  must  be  found  by  means  of 
(64);  and,  in  the  same  manner,  any  other  elements  which  determine 
the  form  of  the  orbit  and  the  position  of  the  body  in  its  orbit,  may 
be  introduced. 

The  partial  differential  coefficients  with  respect  to  the  elements 
having  been  found,  we  have 

do         ^  do         ^  do      .^   ,  do 

cos  r)  diO  =^  cos  rj   -r-  A;^  -\-  COS  rj  -   -  Af  -\-  COS  i?    ,  ,^  i^M^  -\-  COS  ij    ,     A^t, 


dx 


dtp 


dM. 


dii 


dv        r^     ,    drj  dr)  drj  ,      drj        ,r    ,    drj 


dii 


di 


dx 


dtp 


dAL 


dn 


Heuce  we  ol 


DIFFEUENTIAL    FORMULA. 


157 


from  which  it  iippoivM  thut,  by  the  introchiction  of  f^  as  one  of  the 
elt'iiiciits  (»f  the  orbit,  when  the  ^foocentrio  phiws  are  rot'crrod  directly 
to  the  j)huie  of  the  uiiehanjjjed  orl)it  as  the  fiuuhiinental  j)hiiie,  the 
ViU'iiitioii  of  the  {geocentric  longitude  in  reference  to  this  plane  depends 
on  only  four  elements. 

57.  It  remains  now  to  derive  the  formula;  for  finding  tlic  values 
of  Tf  and  d  from  those  of  X  and  (i.  Let  .r^,  ^„,  z^  be  the  geocentric  co- 
onliiiates  of  the  body  referred  to  a  system  in  which  tiie  ecliptic  is 
tlic  plane  of  xy,  the  positive  a.\is  of  x  being  directed  to  the  point 
whose  longitude  is  Si ',  and  let  x^^  y/,  z^  be  the  geocentric  co-ordi- 
nates of  the  l)ody  referred  to  a  system  in  which  the  axis  of  x  remains 
the  same,  l)ut  in  which  the  plane  of  the  unchanged  orbit  is  the  plane 
of  x\j',  then  we  shall  have 


and  also 


Heuce  we  obtain 


.r,  =n  J  cos  /9  cos  {X 

-9.\ 

<- 

=  d  cos  rj  cos  0, 

7/u  =  J  cos  /?  sin  (A 

-«). 

yo'  = 

=  J  cos  yj  sin  0, 

2„  =  J  sin  ,5, 

'o'  = 

=  J  sin  Tj, 

<- 

^0. 

Zq  =  —  ijo  sin  i  -\-  Zq  cos  i. 


cos ij  cos  <?  =  cos /3  cos  (A  —  Q), 

cos  rjsiuO:=  cos  /?  sin  (A  —  Q  )  cos  i  -\-  sin  /?  sin  i, 

sin  J?  =  —  cos  /3  sin  (A  —  J^  )  sin  i  -\-  sin  ,3  cos  ^. 


(80) 


These  equations  correspond  to  tlie  relations  between  the  parts  of  a 
spherical  triangle  of  which  the  sides  are  i,  90°  —  i],  and  90°  —  /9, 
the  angles  opposite  to  90°  —  r^  and  90°  —  /9  being  respectively 
90=  +  (A  —  ft)  and  90°  —  d.  Let  the  other  angle  of  the  triangle  be 
denoted  by  y,  and  we  have 


cos  r}s\nY  =  sin  i  cos  (A 
cos  rj  cos  y  =:  sin  i  sin  (A 


SI), 

ft  )  sin  /?  +  cos  i  cos  ,3. 


(81) 


The  equations  thus  obtained  enable  us  to  determine  ■j,  d,  and  y  from 
/.  and  ^9.  Their  numerical  application  is  facilitated  by  the  intro- 
duction of  auxiliary  angles.     Thus,  if  we  put 


nsiniV=  sin/?, 

n  cos  N  =  cos  /?  sin  (A  —  ft  ), 


(82) 


158 


TIIf:ORETirAL   ASTRONOMY. 


in  which  n  is  always  positive,  we  get 


cos  >?  COS  0  =  COH  (5  COS  (X  —  (J  ), 

co.s  r^  h'\uO  z^  n  cos  ( ^V  —  i), 
sin  )j  =  n  sin  ( ^  —  i), 


(83) 


from  wliich  rj  and  d  may  be  readily  found.     If  wc  also  put 

SI), 


n'  sin  N'  =^  cos  i, 

n'  cos  iV'  =  sin  i  sin  (A 


we  shall  have 


cot iV  =  tan  t  sin  (A  —  Q,), 


cos iV'  ^ ,.       ^ . 


(84) 


(85) 


If  ;-  is  snail,  it  may  be  found  from  the  equation 

sin  i  cos  (A  —  JJ) 


sin/- 


COS); 


(86) 


The  quadrants  in  which  the  angles  sought  must  be  taken,  are  easily 
determined  by  the  relations  of  the  quantities  involved  j  and  the 
accuracy  of  the  numerical  calculation  may  be  checked  as  already 
illustrated  for  similar  cases. 

If  we  apply  Gauss's  analogies  to  the  same  spherical  triangle,  we  gel 

sin  (45°  —  Iri)  sin  (45°  -  ^{0  +  y))  = 

cos (45°  +  ^  (>l  —  a))  sin  (45°  —  A  (/?  +  i)\ 
sin  (45°  —  li)  cos  (45°  —  ^  (fl  +  ;-))  = 

sin  (45°  +  4  (A  —  SJ))  sin  (45°  -  ^  (/?  ~  i)\ 

cos  (45°  —  Aij)  sin  (45°  —  ^^{0  —  y))  =  (87) 

cos  (45°  +  U^  —  S^))  cos  (45°  —  Uji  +  r)). 

cos  (45°  —  irj)  cos  (45°  —  -^{0  —  y))  ^ 

sin  (45°  +  ^  (A  —  J^ ))  cos  (45°  -  ^  (/5  -  0), 

from  w'hich  we  may  derive  rj,  d,  and  y. 

When  the  problem  is  to  determine  the  corrections  to  be  applied  to 
the  elements  of  the  orbit  of  a  heavenly  body,  in  order  to  satisfy 
given  observed  places,  it  is  necessary  to  find  the  expressions  for 
cos;y  Ad  and  ayj  in  terms  of  cos/9  £^?<  and  a/9.  If  we  diiferentiate  the 
first  and  second  of  equations  (80),  regarding  SI  and  i  (which  here 
determine  the  position  of  the  fundamental  plane  adopted)  as  con- 
stant, eliminate  the  terms  containing  dr^  from  the  resulting  equations, 
and  reduce  by  means  of  the  relations  of  the  parts  of  the  spherical 
triangle,  we  get 


NUMEHICAI<   EXAMPLE.  159 

COS  yj  do  =  cos  y  cos  /?  dX  -{■  sin  y  dii. 

Dillcrciitiatiiig  the  last  of  tHumtions  (80),  nml  reducing,  wo  find 

dfj  =  —  siii  Y  co.s  I'i  dX  -\-  cos  y  d,3. 

The  equations  thus  derived  give  the  vuhies  of  the  differential  co- 
(flicients  of  d  and  r^  with  respect  to  X  and  ,9;  and  if  the  ditferencea 
U  and  A;9  arc  small,  we  shall  have 


cos  rj  aOz=  cos  y  cos  /?  aA  -f-  sin  y  ^j3, 
A)j  =  —  sin  /  cos  /S  aA  -f  cos  y  A/?. 


(88) 


The  value  of  y  reciuirod  in  the  application  of  nuuihers  to  these 
c(|U!iti()ns  may  generally  be  derived  with  sufficient  accuracy  from 
(86),  the  algebraic  sign  of  cos^  being  indicated  by  the  second  of 
equations  (81);  and  the  values  of  rj  and  d  required  in  the  calculation 
of  the  diifercntial  coefficients  of  these  quantities  with  respect  to  the 
elements  of  the  orbit,  need  not  be  determined  with  extreme  accuracy. 

58.  Example. — Since  the  si)hcrical  co-ordinates  which  are  fur- 
uisliod  directly  by  observation  arc  the  right  ascension  and  declina- 
tion, the  formulro  will  be  most  frequently  required  in  the  form  for 
tiiuling  ;y  and  d  from  a  and  u.  For  this  purpose,  it  is  only  necessary 
to  write  a  and  d  in  place  of  A  and  fi,  respectively,  and  also  SI',  i', 
w', '/',  and  w'  in  place  of  SI,  i,  <o,  1,  and  k,  in  the  equations  which 
hiivc  been  derived  for  the  determination  of  "q  and  </,  and  for  the 
differential  coefficients  of  these  quantities  with  respect  to  the  elements 
of  the  orbit. 

To  illustrate  this  clearly,  let  it  be  required  to  find  the  expressions 
for  cos  t}  A^  and  A;y  in  terms  of  the  variations  of  the  elements  in  the 
case  of  the  example  already  given ;  for  which  we  have 

(«'  =  50°  10'  7".29,        ^'==353'  45'  35".87,        i'  =  19°  26'  25".76. 

These  are  the  elements  which  determine  the  position  of  the  orbit  of 
Eunjnome  @,  referred  to  the  mean  equinox  and  equator  of  1865.0. 
We  have,  further, 


log/ =0.62946, 
i^=339°14'0", 


log  g  =  0.34593,  log  h  =  2.97759, 

G  =  350°  11'  16",         H=  14°  30'  48", 
w'  =  179°13'58". 


In  the  first  place,  we  compute  ^,  d,  and  y  by  means  of  the  formulro 


'A'i 


160 


THEORETICAL  ASTUOKOMY. 


(83)  and  (85),  or  by  means  of  (87),  writing  a,  d,  ^',  and  i'  instead 
of  /,  /J,  SI,  and  i,  respectively.     Hence  we  obtain 


0  =  lHS°'ird", 


yj^  —  r  oW  28", 


19°  17' 7". 


Since  the  equator  is  lierc  considered  as  the  fundamental  plane,  the 
longitude  0  is  jueasured  on  the  equator  from  the  pi.'ce  of  the  asc«!n(]- 
ing  notle  of  the  orbit  on  tliis  plane.  The  values  o.'"  the  differential 
coefficients  arc  then  ibuud  by  means  of  the  formula} 


cos.^^^,^0, 
do 

do 


COS  5j  -J—  =  •-  cos  (0  —  u'), 


dO 


cos  5}  —,-  =:=  —  cos  {0  —  u'  —  F), 


cos 


df 

cos  5j  — ,--  -—  —  cos  {0  —  U  —  H), 

which  give 

do 

do 

cost;  —V-,-  -r-.  +  1.5051, 


cos  >j  --,---  ^-^-  -f  2.0978, 


eoiV  // 


do 
d<p 

do 

dM, 

do 


4- 1.1922, 


cos  fj  -^—  r=  J-  538.00, 


drj  r  .    .,         , 

-j-—f  =  —  —  cos  t]  sui  %  cos  u , 

—pr  =  +  -  cos  ij  siL  u, 

Aj>^  r=.  _  4  sin  ri  sin  {0  —  u'  —  F), 
d<p  J 

dri 
d/J. 


'-r  sin  rj  sin  {0  —  ii'  —  G), 
-  —  sin  Tj  sin  {6  —  u'  —  //), 


dv, 
dSl' 


+  0.6072, 


-]L.  -^  +  0.0204, 
di 

~P-  ==  4-  0.0086, 


df] 
d<p 

dM, 

dfj. 


^  +  0.0422, 
-=  +  0.0143, 
-^-1.71. 


Therefore,  the  equations  for  cos  jy  aO  and  a;^  become 

cos  r,  AO  .:=..  -{-  1.5051  A;/'  -f  2.0978  A<p  +  1.1922  AiJ/;,  +  538.00  A/x, 
A/j  =  -f  0.0086  A/  +  0.04'»  sif  4-  0.0143  ^1%  —  1.71  A/x 
+  0.5072  aS^'  +  0.0204  Ai'. 

II"  we  assign  to  the  elements  of  the  orbit  the  variations 


DIFFERENTIAL   FORMULA. 


161 


WO 


c^<P  =  +  10". 
have 


AS2'=r_14".12, 


AM 


10", 


^1':=.  —  8".86, 
A/i  ^  +  0".01, 


A/  ^  Aio'  -f  cos  i'  A  a'  ==:  —  19".96 ; 

ami  the  pi'cccfliiig  equations  give 

cos  >j  Afl  =  +  8".24,  Ai;  z^  —  6".96. 

With  tlic  same  values  of  aw',  a^',  &c.,  we  have  already  found 

cos  S  Aa  =  +  5".47,  A<5  ^  —  9".29, 

which,  by  means  of  the  equations  (88),  writing  a  and  8  in  place  of 
).  and  ,?,  give 


COS1jA<?r=:  +  8".23, 


A1J  .^  —  6".96. 


59.  In  special  cases,  in  which  the  differences  between  the  calcu- 
lated and  the  observed  values  of  two  spherical  co-ordinates  are  given, 
and  the  corrections  to  be  aj)plied  to  the  a :^•Jumed  elements  are  sought, 
ir  niav  become  necessary,  on  account  of  dithculties  to  be  encoinitered 
in  the  solution  of  the  equations  of  condition,  to  introduce  other  ele- 
ments of  the  orbit  of  the  body.  The  relation  of  the  elements  chosen 
io  thosi'  commonly  used  will  serve,  witliout  presenting  any  dithculty, 
i'or  the  transformation  of  the  equations  into  a  fornv  adapted  to  the 
special  ease.  Thus,  in  the  ease  of  the  elenu'uts  which  determine  the 
form  of  the  orbit,  we  may  use  a  or  log  a  instead  of  n,  and  the 
pq  nation 


3 


gives 


d/i 


^  a 


-  2  J  (I  log  a, 


(89) 


ii.  vhich  /,,  is  the  modulus  of  the  system  of  logarithms.     Tlierefore, 
the  coefficient  of  A/i  is  transformed  into  that  of  a  logw/  by  multiply- 

ing'  it  by  —  J--;  and  if  the  unit  of  the  ?».th  decimal  place  of  the  loga- 

nthms  is  taken  as  the  unit  of  a  log  a,  the  coefficient  must  be  also 
niultipHcd  by  10".  The  homogeneity  of  the  equation  is  not  disturbed, 
since  n  is  iiere  supposed  to  be  expressed  in  seconds. 
If  we  introduce  log^^  as  one  of  the  elements,  from  the  equation 


p  =  a  cos*  fp 
11 


162 


we  get 


or 


TIIEOBETICAI,  ASTRONOMY. 


d  log^  =  —  i  -  «^At  —  2^>o  tau  <p  dip, 


dii  =  —  I  Y  d  logj)  —  o/Jt  tau  <p  d(p. 


(90) 


HoiK'o  it  appears  that  the  coefficients  of  a  logj)  are  the  same  as  those 
of  A  log  fl,  but  since  />  is  also  a  function  of  f,  the  coefficients  of  Ap 

are  changed ;  and  if  we  denote  by  cos  o  I  y-  I  and  I  ,—  I  the  values  of 

the  partial  differential  coefficients  when  the  element  fjt  is  used  in  con- 
nection with  tp,  wo  shall  have,  for  the  case  under  consideration, 


.do. 

cos  o  1— 

d<p 

dH 

dtp 


cos 


.1  da  \        ,,  //  .da 

3  ._  3 '_  tan  <p  cosd-^-, 

\d(p  I         s  d[i 

ldd\       _/..  dd 


in  which  s  =  2062G4".8.     If  the  values  of  the  diffisrential  coefficients 

with  respect  to  n  and  f  have  not  already  been  found,  it  will  be  ad- 

,          ,          ^  dr    dv        dr            ,       dv      , 
vantageous  to  compute  the  values  oi  -.— ,  -f—,  -n «  and    ,  , by 

'^  '■  dip   dtp    d  log^)  d  logj>    ■' 

means  of  the  expressions  which  may  be  derived  by  substituting  in 
the  equations  (15)  the  value  of  d/z  given  by  (90),  and  then  we  may 

compute  directly  the  values  of  cos  5-^—,  cos  8  -t-. ,  -t->  and  -n — -• 

*  "^  dip  d  logp   dip  d  log;) 

In  place  of  31^,  it  is  often  convenient  to  introduce  L^,  the  mean 

longitude  for  the  epoch;  and  since 


we  have 


dL^  =  dM^  -\-dx=^  dM„  +  d'o-\-  dSl, 

and,  when  ;f  is  used, 

dLg  =  dMf,  -\-  dx-\-  {l—  cos  i)  r? ft . 

Instead  of  the  elements  ft  and  i  which  indicate  the  position  of  the 
plane  of  the  orbit,  we  may  use 


b  =  sin  %  sin  ft, 


c  =  sini  cos  ft, 


and  the  expressions  for  the  relations  between  the  differentials  of  h 
and  e  and  those  of  i  and  ft  are  easily  derived.  The  cosines  of  the 
angles  which  the  line  of  apsides  or  any  other  line  in  the  orbit  makes 
with  the  three  co-ordinate  axes,  may  also  be  taken  as  elements  of  the 


DIFFERENTIAL   FORMULAE. 


163 


orbit  in  the  formation  of  the  equations  for  tlie  variation  of  the  geo- 
centric phice. 

60.  The  equations  (48),  by  writing  I  and  h  in  pUice  of  X  and  /9, 
respectively,  will  give  tiie  values  of  the  differential  coefficients  of 
the  heliocentric  longitude  and  latitude  with  respect  to  x,  y,  and  z. 
Combining  these  with  the  ex})ressions  for  the  differential  coefficients 
of  the  heliocentric  co-ordinates  with  respect  to  the  elements  of  tiie 
orbit,  we  obtain  the  values  of  cos  b  \l  and  a6  in  terms  of  the  varia- 
tions of  the  elements. 

The  equations  for  dx,  dy,  and  dz  in  terms  of  du,  dSl,  and  di,  may 
also  be  used  to  determine  the  corrections  to  be  applied  to  the  co-or- 
dinates in  order  to  reduce  them  from  the  ecliptic  and  mean  equinox 
of  one  epoch  to  those  of  another,  or  to  the  apparent  equinox  of  the 
date.     In  this  case,  we  have 

dv  ==(?;:  —  dSl. 

When  the  auxiliary  constants  A,  B,  a,  h,  etc,  are  introduced,  to 
find  the  variations  of  these  arising  from  the  variations  assigned  to 
the  elements,  we  have,  from  the  equations  (99),, 

cot  A  =  —  tan  Q,  cos  r, 

cot  iS  =  cot  Q  cos  i  —  sin  i  cosec  Q,  tan  e, 

cot  C  =  cot  SI  cos  /  -f  sin  i  cosec  $^  cot  e, 

in  which  i  may  have  any  value  from  0°  to  180°.  If  we  differentiate 
these,  regarding  all  the  quantities  involved  as  variable,  and  reduce 
by  means  of  the  values  of  sin  a,  sin  6,  and  sin  c,  we  get 


dA  = 
dB  = 

+ 


cosi    ,^        sinA    .    ^    .    .  ,, 
— j—dSl ;t,.  ;r  ^'^  o6  sni  i  at, 


8m*a 

cose 

sin'^  b 

sin  B 


sma 
(cos  i  cos  £  —  sin  i  sin  e  cos  JJ  )  fZj^ 


r      r^    •    •           f         .  .     .  J.  ,  smism  Si  , 
•    .  (cos  Si  sm  t  cos  £  A-  cos  i  sm  e)  di  A .  -,-; — -  as. 


sme 


dC=  —.-—r-  (cos i  sin  e  4-  sm  i  cos  e  cos  SI)  dQ 

sinC,       _    ,    .  .                 .         ST.,   sin  i"  sin  Si  , 
"r  „;„  „■  (cos  SJ  sni  I  sm  £  —  cos  t  cos  £)  di  -\ .—^ ds  • 


smc 


sm'c 


and  these,  by  means  of  (lOl),,  reduce  to 


164  THEORETICAL  ASTRONOMY. 

dA  =  -  v-V,—  dQ  —  sin  A  cot  a  di, 
sill'  a 

,„       cose  cose  .  ,  coso    , 

uB  =  —  .    : , —  a  Si  —  sill  B  cot  0  di  ~\ — -^rr  de, 
sill' 6  sin' 6 

,  ^           sin  s  cos  b 
dU=^ 


(91) 


dSl  —  sin  C  cot  c  di  +  -r-V-  ds. 


Let  us  now  differentiate  the  equations  (lOl)i,  using  only  the  upper 
sign,  and  the  result  is 

da=:  —  sin  i  sin  A  dQ,  4-  cos  ^  di, 

db  =^  —  sin  i  sin  B  dSl  -\-  cos  B  di  -\-  cos  c  cosec  b  ds, 

de  =  —  sin  /  sin  C  dQ,  -\-  cos  C  di  —  cos  b  cosec  c  ds. 

If  we  multiply  the  first  of  these  equations  by  cot  a,  the  second  hy 
cot 6,  and  the  third  by  cote,  and  denote  by  X^  the  modulus  of  the 
system  of  logaritlinis,  we  get 

d  log  sin  a  =  —  /io  sin  /  cot  a  sin  Ad^  -{-  >-q  cot  a  cos  A  di, 

d  log  sin b  =  —  ^q  sin  /  cot  b  sin  B  dQ,  +  ^o  cot6  eos  Bdi -\-  A^ ^-.^j^ —  ds, 


cZ  log  sinc=  —  Ajsint  cote  sin  CdQ,  -|- -'•o  cote  cos  CcZi — \ 


sin'  b 

cos  b  cos  c 
sin'  c 


de. 


The  equations  (91)  and  (92)  furnish  the  differential  coefficients  of 
A,  B,  C,  log  sill «,  &e.  with  respect  to  SI,  i,  and  e;  and  if  the  varia- 
tions assigned  to  Q,,  i,  and  £  are  so  small  that  their  squares  may  l)e 
neglected,  the  same  equations,  writing  aA,  aQ,,  a/,  &c.  instead  of 
the  differentials,  give  the  variations  of  the  auxiliary  constants.  In 
the  case  of  ecpiations  (92),  if  the  variations  of  Q,,  /,  and  c  are  ex- 
pressed in  seconds,  each  term  of  tlie  second  member  must  be  divided 
by  20G2G4.8,  and  if  the  variations  of  log  sin  a,  log  sin  6,  and  log  sine 
are  required  in  units  of  the  mt\i  decimal  place  of  the  logarithms,  each 
term  of  the  second  member  must  also  be  divided  bv  10'". 

If  we  differentiate  the  equations  (81)i,  and  reduce  by  means  of  tlio 
same  equations,  we  easily  find 


cos  b  dl  =^  cc  i  sec  b  da  +  cos  b  dfl  —  sin  b  cos  (I  —  SI)  di, 
db  =  sin i  cos (l  —  SI)  du  +  sin (Z  —  SI)  di, 


(93) 


which  determine  the  relations  between  the  variations  of  the  elements 
of  the  orl)it  and  those  of  the  heliocentrif;  longitude  and  latitude. 
Jjy  differentiating  the  equations  (88),,  neglecting  the  latitude  of 


DIFFERENTIAL   FORMULAE. 


165 


tlie  sun,  and  considering  A,  /?,  J,  and  ©  as  variables,  we  derive,  after 
reduction, 


cos  /3  cU  =  _  cos  (A  —  O  )  f^O , 


d^. 


E 


(94) 


sin/Jsin(;.  —  0)f/0, 


which  determine  the  variation  of  tlie  geocentric  latitude  and  longitude 
arising  from  an  increment  assigned  to  tlie  longitude  of  the  sun.  It 
apjioars,  therefore,  that  an  error  in  the  longitude  of  the  sun  will 
produce  the  greatest  error  in  tlie  computed  geocentric  longitude  of  a 
lieavenly  body  when  the  body  is  in  opposition. 


166 


THEORETICAL   ASTRONOMY. 


CHAPTER  III. 


and  in  declin 
Thus,  we  hiv 


INVESTIGATION  OP  FORMULA  FOR  COMPrTIXO  THE  ORBIT  OF  A  TOMET  MOVING  IN 
A  PARABOLA,  AND  FOR  CORRECTINO  AIU'KOXIMATE  ELEMENTS  BY  THE  VARIATION 
OF  THE  GEOCENTRIC   DISTANCE. 

61.  TiiK  observed  spherical  co-ordinates  of  the  place  of  a  heavenly 
body  furnish  each  one  equation  of  condition  for  the  correction  of  the 
elements  of  its  orbit  approximately  known,  and  similat'ly  for  tlie 
determination  of  the  elements  in  the  ease  of  an  orbit  wholly  unknown; 
and  since  there  are  six  dements,  neglecting  the  mass, — whicli  nnist 
always  be  done  in  the  first  approximation,  the  perturbations  not 
being  considered, — three  complete  observations  will  furnish  the  six 
equations  necessary  for  finding  tlicse  unknown  quantities.  Hence, 
the  data  required  for  the  determination  of  the  orbit  of  a  lieavenly 
body  are  three  complete  observations,  namely,  thr>  e  observed  longi- 
tudes and  the  corresponding  latitudes,  or  any  other  spherical  co- 
ordinates which  completely  determine  three  places  of  the  Ixxly  as 
seen  from  the  earth.  Since  these  observations  are  given  as  made  at 
some  point  or  at  diiferent  points  on  the  earth's  surface,  it  becomes 
necessary  in  the  first  place  to  apply  the  corrections  for  parallax.  In 
the  case  of  a  body  whose  orbit  is  wholly  unknown,  it  is  impossible 
to  apply  the  coiTcction  for  parallax  directly  to  the  place  of  the  body; 
but  an  equivalent  correction  may  be  a])plied  to  the  places  of  the 
earth,  acctyding  to  the  formuhe  which  will  be  given  in  the  next 
chapt«.r.  However,  in  the  first  determination  of  approximate  elc- 
nient^i  of  the  orbit  of  a  comet,  it  will  be  sufficient  to  neglect  entirely 
tlic  coiTcction  for  parallax.  The  uncertainty  of  the  observed  places 
of  these  bodies  is  so  much  greater  than  in  the  case  of  Avell-definod 
objects  like  the  planets,  and  tlic  intervals  between  the  observations 
whitili  will  be  generally  employed  in  the  first  determination  of  tho 
orbit  will  be  so  small,  that  an  attempt  to  represent  the  observed  places 
with  extreme  accuracy  Avill  be  superfluous. 

Wheii  aJ3proximate  elements  have  been  derived,  wc  may  rind  the 
distances  of  the  comet  from  the  earth  corresponding  to  the  three 
observed  places,  and  lience  determine  the  parallax  in  right  ascension 


in  which  a  i 
of  the  comet 
of  observati( 
observation, 
equatorial  n 
sun. 

In  order  t 
place  by  met; 
tion  must  a 
time  of  ob» 
but  if  J  is  n( 
in  the  first  a 

The  transi 
into  latitude 
which  may  I 
and  writing 


and  also 


which  will 

Since  cos/9 
tho  same  si 
taken. 

G2.  As  so 
have  been  ej 
to  the  same 


DETERMINATION   OF  AN   ORBIT. 


167 


and  in  declination  for  each  observation  by  means  of  the  usual  formulse. 
Thus,  we  have 

■Kp  COS  <p'   sin  (o  —  0) 


Aa  =  — 


C08  0 


tan>'=: 

A<S  = 


J 
tan  <p' 

COS(a^^^:^^' 

-p  sin  <p'    sin  (y  —  8) 
J  sin  Y      ' 


iu  which  a  is  the  right  ascension,  d  the  declination,  J  the  distance 
of  the  comet  from  the  earth,  <p'  the  geocentric  latitude  of  the  place 
of  observation,  0  the  sidereal  time  corresponding  to  the  time  of 
observation,  p  the  radius  of  the  earth  expressed  in  parts  of  the 
equatorial  radius,  and  r:  the  equatorial  horizontal  parallax  of  the 
sun. 

In  order  to  obtain  the  most  accurate  representation  of  the  observed 
place  by  means  of  the  elements  computed,  the  correction  for  aberra- 
tion must  also  be  applied.  When  the  distance  J  is  known,  the 
time  of  observation  may  bo  corrected  for  the  time  of  aberration; 
but  if  J  is  not  approximately  known,  this  correction  may  be  neglected 
in  the  first  approximation. 

The  transformation  of  the  observed  right  ascension  and  declination 
into  latitude  and  longitude  is  cftected  by  means  of  the  equations 
wliich  may  be  derived  from  (92),  by  interchanging  a  and  /,  o  and  ^9, 
and  writing  — e  instead  of  e.     Thus,  we  have 


and  also 


^      AT     tan  ^ 
tan  N=  -. — , 

sma 

cos  (N —  s) 
cosN 
tan  ;5  =  tan  (iV  —  e  )  sin  ^, 


tan  A 


tana, 


^1) 


cos  (N —  e) cos ,?  sin  A 

cos  N  cos  d  sin  tt ' 


which  Avill  serve  to  check  the  numerical  calculation  of  A  and  /9. 
Since  cos  ^9  and  cos<J  are  always  positive,  cos/  and  cos  a  nuist  have 
tlio  same  sign,  thus  determining  the  quadrant  iu  which  ?.  is  to  be 
taken. 

G2.  As  soon  as  these  prelin.inary  corrections  and  transformations 
have  been  effected,  and  the  times  of  observation  have  been  reduced 
to  the  same  meridian,  the  longitudes  having  been  reduced  to  the 


168 


THEORETICAL   ASTRONOMY. 


sanio  equinox,  wc  are  prepared  to  proceed  with  the  determination  of 
the  elements  of  the  orbit.  For  this  purpose,  let  t,  t',  t"  be  the  times 
of  observation,  »•,  r\  r"  the  radii-vectores  of  the  body,  and  «,  u',  n" 
the  corresponding  arguments  of  the  latitude,  R,  R',  R"  the  distances 
of  the  earth  from  the  sun,  and  O,  O',  O"  the  longitudes  of  the  sun 
con'esponding  to  these  times. 

Let  [rr']  denote  double  the  area  of  the  triangle  formed  between 
the  radii-vectores  r,  r'  and  the  chord  of  the  orbit  between  the  corre- 
sponding places  of  the  body,  and  similarly  for  the  other  triangles 
thus  formed.  The  angle  at  the  sun  in  this  triangle  is  the  dift'erence 
between  the  corresponding  arguments  of  the  latitude,  and  we  shall 
have 

[rr']   =  rr'  sin  (it'  —  «)> 

[n-"]  =  rr"  sin  («"  -  u),  (2) 

[,•',•"]  =.  r'r"  sin  («"  -  «'), 

If  wc  designate  by  x,  y,  z,  x',  y',  z',  x",  y",  z"  the  heliocentric  co- 
ordinates of  the  body  at  the  times  t,  t',  and  t",  wo  shall  have 

x'  =  r  sin  a  sin  (A  +  u), 
x'  =z  r'  sin  a  sin  (A  -f-  «')> 
x"  =  r"  sin  a  sin  (^1  -|-  it"), 

in  which  a  and  A  are  auxiliary  constants  which  are  functions  of  the 
elements  il  and  /,  and  these  elements  may  refer  to  any  fundamental 
plane  whatever.  If  we  multiply  ^  3  first  of  these  equations  by 
sin  {u"  —  ?/'),  the  second  by  —  sia  {u"  —  u),  and  the  third  by 
sin  (it'  —  u),  and  add  the  products,  we  find,  after  rediction, 


a  comet,  u 


X 


X 


sin  {u"  —  xi') ,  sin  (it" 


«)  +  -^  sin  (it'  ■ 

T 


it)  =  0, 


which,  by  introducing  the  values  of  [/*;•'],  [rr"],  r.nd  [r'  r"],  becomes 
[rV']  X  —  Irr"]  x'  +  [r/]  x"  =  0. 
[rV] 

we  get 


-V] 

,."T  ' 


If  WC  put 

nx  —  xf  +  n"x"  =  0 
In  precisely  the  same  manner,  we  find 


"   -\rr"^ 


(3) 
(4) 


nt/-i/'  +  «'y'  =  0, 
nz  —  z'^  n"z"  =  0. 


(5) 


DETERMINATION  OF   AN   ORBIT. 


169 


Since  the  coefficients  in  tliesc  equations  arc  indepcnrlcnt  of  the  posi- 
tions of  the  co-ordinate  planes,  except  that  the  origin  is  at  the  centre 
of  the  sun,  it  is  evident  tliat  the  three  equations  are  i(U'nti<'al,  and 
express  simply  the  condition  that  the  plane  of  the  orbit  passes  through 
the  centre  of  the  sun  ;  and  the  last  two  might  have  been  derived 
from  the  first  by  writing  successively  (/  and  z  in  place  of  .r. 

Let  I,  )J,  X"  be  the  three  observed  longitudes,  /9,  /9',  [i"  the  corre- 
sponding latitudes,  and  J,  J',  J"  the  distances  of  the  body  from  the 
eiu'th;  and  let 

J  cos  /?  =  p,  A'  cos  ;/  =  p',  A"  cos  ,5"  =  p'\ 

which  are  called  curtate  distances.     Then  we  shall  have 


x  =  p  cos  A  —  B  cos  O , 
y  =^  p  sin  X  —  E  sin  O , 
z  =^p  tan  i3, 


x'  —-  p'  cos  A'  —  R  cos  ©', 
y'=/,'siuA'— iJ'sinO', 
z'  =  p  tan  [i', 


x"  =  p"  COS ?."  — Ji"  cos  Q", 
y"^p"mi)."  —  R"smQ", 
2"  -=  p"  tan  <5", 

in  wliich  the  latitude  of  the  sun  is  neglected.  The  data  may  be  so 
transformed  that  the  latitude  of  the  sun  becomes  0,  as  will  be  ex- 
plained in  the  next  chapter ;  but  in  the  com])utation  of  the  orbit  of 
a  comet,  in  which  this  preliminary  reduction  has  not  been  made,  it 
will  be  unnecessary  to  consider  this  latitude  which  never  exceeds  1", 
while  its  introduction  into  the  formuhe  w'ould  unnecessarily  com- 
plicate some  of  those  which  will  be  derived.  If  we  substitute  these 
values  of  x,  x',  &c.  in  the  equations  (4)  and  (5),  they  become 


0  :=  ji  (jo  cos  A  —  i?  cos  O  )  —  ip'  cos  A'  —  R'  cos  ©') 

+  n"(/>"cosA"  — ii"cosO"), 
0  =  «  (/)  sin  A  —  i?  sin  O  )  —  (/>'  sin  A'  —  R'  sin  ©') 

+  n"(/f>"sinA"  — irsinO"), 
0  =  np  tan  ^  —  p'  tan  ,5'  +  w'V"  tan  ,S". 


(6) 


These  equations  simply  satisfy  the  condition  that  the  plane  of  the 
orbit  passes  through  the  centre  of  the  sun,  and  they  only  become 
distinct  or  independent  of  each  other  when  n  and  n"  are  expressed 
in  functions  of  the  time,  so  as  to  satisfy  the  conditions  of  undisturbed 
motion  in  accordance  with  the  law  of  gravitation.  Further,  they 
involve  five  imknown  quantities  in  the  case  of  an  orbit  wholly 
unknown,  namely,  n,  n",  p,  p',  and  p" ;  and  if  the  values  of  n  and 
n"  are  first  found,  they  will  be  sufficient  to  determine  p,  p',  and  p". 


170 


TIIEOTIETICAI.   ASTEONOMY. 


The  determination,  however,  of  n  and  n"  to  a  snflfioient  degree  of 
acniracv,  bv  means  of  the  intervils  of  time  between  the  ol)servation8, 
requires  that  //  should  bo  approximately  known,  and  henee,  in 
general,  it  will  become  necessary  to  derive  first  the  values  of  n,  >/', 
and  f*' ;  after  which  those  of  ft  and  p"  may  be  found  from  ecpiations 
(6)  by  elimination.  But  since  the  number  of  equations  will  then 
exceed  the  nundier  of  unknown  quantities,  we  may  combine  them  in 
such  a  manner  as  will  diminisli,  in  the  greatest  degree  possible,  tlio 
effect  of  the  errors  of  the  observations.  In  special  cases  in  which 
the  conditions  of  the  problem  are  sueli  that  when  the  ratio  of  two 
curtate  distances  is  known,  the  distances  themselves  may  be  deter- 
mined, the  elimination  must  be  so  performed,  as  to  give  this  ratio 
with  the  greatest  accuracy  practicable. 

63.  If,  in  the  first  and  second  of  equations  (6),  we  change  the 
direction  of  the  axis  of  x  from  the  vernal  equim  \-  to  the  place  of  the 
sun  at  the  time  <',  and  again  in  the  second,  fp"!  the  equinox  to  the 
second  place  of  the  body,  we  must  diminish  tl  longitudes  in  these 
equations  by  the  angle  through  which  the  axis  of  x  hais  been  moved, 
and  we  shall  have 


0  =  nip  cos(>l  —  Q')—Rco3(Q'—  Q))  -  (r'  cos(k'—  ©')  —  E') 
+  7i"(p"  cos(;."—  0')  -  Ji"  cos(0"-  ©')), 

0  =  n(psm(X  —  O')-{-Emi(Q'—Q))—r'sm(k'—Q') 

+  h"  ((>"  sin  (A"  -  O ' )  -  li"  sin  ( O  "  -  O ')),  (7) 

0  =  11  (f>  sin (X'  —  X)-\-lism(Q— )! ))  —  li'  sin  ( © '  —  A') 
—  n"  (/."  sin  (A"  —  /')  —  R"  sin  (©"  —  k')\ 

0  =  np  tan  ,3  —  p'  tan  ,5'  +  n"p"  tan  /5". 

If  we  multiply  the  second  of  these   equations   by  tan  ^9',  and   the 
fourth  by  —  sin(// —  ©'),  and  add  the  products,  we  get 

0  =  7iVaan/5'sin(;."—  ©')  —  tan,'/' sin(/l'-  ©')) 
—  ?i"i2" sin  (©"—©')  tan  ,/+?)/>  (tan /3' sin  (A  —  ©')  — tan  ,3 sin  (/'—©')) 

+  nR  sin  (©'—©)  tan  /3'.  (8) 

Let  us  now  denote  double  the  area  of  the  triangle  formed  by  the 
sun  and  two  places  of  the  earth  corresponding  to  R  and  R'  by  [i?-R'], 
and  we  shall  have 

iRR''\  =  RR'  sin(©'—  ©), 
and  similarly 

IRR"]  =  RR"  sin  ( ©  "  —  © ), 
iR'R''-^  ^  R'R"  sin (©"-©')• 


OUBIT  OF   A   HEAVENLY   BODY. 


171 


Then,  if  we  put 


N: 


lli'Ii"] 


,11-,  > 


we  obtain 


[7;A"'] 


N"  = 


[/»'A"] 


.» /,'"i ' 


UiJi"] 


(9) 


i?"siu(0"-0')  =  i2  8in(0'-  O)  ..r, 


iV" 


Stil)stitnting  this  in  the  equation  (8),  and  clivitling  by  the  cocffioicnt 
of  //'',  the  result  is 


;" 


tan  fi'  m\ (X  —  Q')  —  tan  ,?  sin  (/.'  —  O ') 


u"    tan  ,5"  sin  {X'  _  © ' )  —  tan  ,i'  sin  (A"  —  ©') 

X\  AsinfO'— 0)tani9' 


"^\«"       iV")tan/i"gin(A' 


©')  — tan/5'8in(r— ©')' 

Let  ns  also  put 

, ,,  _  jtan /i' sin  (A  —  O ' )  —  tan  <?  sin  (■/ —  vj' ) 
~"  taTT ^FSii  (A'  _■  o')rzrtan  /5'  sin  (P^^  ©'7 


8in(©'—  ©)tan;^y 


V 


(10) 


tan  ,3"  sin  (A'  —  ©')  —  tan ,'/  sin  (A"  —  ©')' 
and  the  preceding  equation  reduces  to 

X 


^'-|^^>  +  (y^-|^)^"^- 


(11) 


We  may  transform  the  values  of  M'  and  31"  so  as  to  be  bettt^r 

adapted  to  logarithmic  calculation  with  the  ordinary  tables.     Thus, 

if  w'  denotes  the  inclination  to  the  ecliptic  of  a  great  circle  passing 

through  the  second  place  of  the  comet  and  the  second  place  of  the 

sun,  the  longitude  of  its  ascending  node  will  be  ©',  and  we  shall 

have 

sin  (A'  —  ©')  tan  w'  =  tan  /5'.  (12) 

Let  j3^,  i%"  be  the  latitudes  of  the  points  of  this  circle  corresponding 
to  the  longitudes  k  and  /",  and  we  have,  also, 


tan  /?o  =  sin  (A  —  ©')  tan  ^v', 
tan  /V  =  sin  (A"  —  ©')  tan  w'. 


(13) 


Substituting  these  values  for  tan/3',  sin(^ — ©')  and  sin  (A" — ©') 
in  the  expressions  for  31'  and  31",  and  reducing,  they  become 


31' 


sin  (/?„  —  /9)     cos  /5"  cos  ,V' 


sin  ((3"  —  fi^')      cos/?(,cos/3 
M"  =  tan  w'  sm  (©' -  ©)  -^, — ^. 


(14) 


172 


TIIEOUKTICAL   ASTUON'OMY. 


n 


When  tlio  value  of   ",   has  boon  fouiul,  ^(Hiufion  (11)  will  ^ivc  the 

relation  hotwot'ii  p  and  />"  in  terms  of  kiiown  quantiticH.  It  is  evi- 
dent, however,  from  etjuations  (14),  that  when  the  apparent  [>ath  of 
the  eomet  is  in  a  piano  passing  through  the  second  plaee  of  the 

sun,  since,  in  this  case,  1^  ~- 1%  and  ["t" —-  ft,^" ,  we  shall  have  M'- 


0 

d 


and  M"  ~  oo.  In  this  ease,  therefore,  and  also  when  i%  —  /9  and 
[i"  —  /9|,"  are  very  nearly  0,  we  nuist  have  recourse  to  some  other 
ccpiation  which  may  be  derived  from  the  equations  (7),  and  which 
docs  not  involve  this  indetcrniination. 

It  will  he  observed,  also,  that  if,  at  the  time  of  the  middle  obser- 
vation, the  comet  is  in  opposition  or  conjunction  with  the  sun,  the 
values  of  M'  and  M"  as  given  by  e(|uation  (14)  will  be  indeter- 
minate in  form,  but  that  the  original  equations  (10)  will  give  tlio 
values  of  these  quantities  provided  that  the  ap})arent  ])ath  of  tlio 
comet  is  uot  in  a  great  circle  passing  tlu'ough  the  tjccond  place  of  the 
sun.     These  values  are 


M'=-- 


sinU-Q') 
sin  (/'—©')' 


M"=- 


sin  (Q^—Q) 


Hence  it  appeal's  that  whenever  the  aj>parent  path  of  the  body  is 
nearly  in  a  plane  passing  through  the  place  of  the  sun  at  the  time  of 
the  middle  observation,  the  errors  of  observation  will  have  great 
influence  in  vitiating  the  resulting  values  of  M'  and  M" ;  and  to 
obviate  the  difficulties  thus  encountered,  we  obtain  from  the  third  of 
equations  (7)  the  following  value  of  //' : — 


Bin(A'  — ;i) 


+ 


-^i?sin(0-A') 


siaCA"  — A') 
i^ii'sinCO'  — /)+i2"sin(0"  — A')^^^^ 


sin  (/"  —  A') 


We  may  also  eliminate  p  between  the  first  and  fourth  of  equa- 
tions (7).     If  we  muliiply  the  first  by  tan  ^9',  and  the  second  by 

—  cos  (A' —  ©'),  and  add  the  products,  we  obtain 

0  =  ft'V  (tan  /3'  ('OS  (A"  —  ©')  —  tan  /?"  cos  (A'  —  ©')) 

—  n"i2"tan/5'cos(©"— ©')+«/>(tan/5'cos(A— ©')  — tan/5cos(A'— 0')) 

—  nE  tan  [n'  cos  (©'—©)  +  ii'  tan  /S', 

from  which  we  derive 


OIUUT   OF    A    IIKAN'KNr.Y    liODV 


173 


7." 


tnn  ,y  009  {X 


O ' )  —  tnn  ,i^'08  (A'  —_0') 
tan  ,J"  cow  ( k'  —O')  —  tanji'  cort'i  k"  ~Q ')  (HI) 

7i"'  tan  fi'  coh  (0"—  © ')  +  -^i7  It  tan ,'/  eoa  ( © '—  O  )  —  J,  A"  tan  ,5' 
tan/i"co8U'— ©')  —  tan,'j' coa(-i"— "©')  * 

Let  u.s  now  (U'note  by  I'  the  int'Ilnation  to  the  cclijitio  of  a  p;r('nt 
circlt'  passin}^  tliroiij^h  the  second  place  of  the  comet  and  that  point 
of  tlic  ecli[)ti(^  wluKse  longitude  is  ©' —  00°,  which  will  therefore  be 
the  longitude  of  its  ascending  node,  and  we  shall  iiave 


COS  (A'—  ©')tan7': 


tan  ,y  \ 


(17) 


and,  if  we  designate  by  /9,  and  /5„  the  latitudes  of  tlie  points  of  this 
circle  corresponding  to  the  longitudes  ).  and  /",  we  shall  also  have 


tan  p,  —  cos  (A  —  ©')  tan  I', 
tan  fl„ ~  cos  (A"  —  ©')  tan  /'. 

Introducing  these  values  into  equation  (IG),  it  reduces  to 

„__      11       sin(,J,  —  /J)      cos /5"  COS/?,, 
''  ~  *"  «"  '  sin"(/V'^^* J  '  "cos7"eos /37 


(18) 


tan /' cos /5"  COS/?,, 


n 


sin(/y 


,'/ 


/5„) 


R"  cos  CO"—  ©')  +  -7-Rcos(0'—  ©) 


n 


(19) 
AM 
n    I 


from  which  it  appears  that  this  equation  becomes  indeterminate  when 
the  apparent  path  of  the  body  is  in  a  plane  passing  through  that 
point  of  the  ecliptic  whose  longitude  is  equal  to  the  longitude  of  the 
si'cund  place  of  the  sun  diminished  by  i)0°.  In  this  case  we  may  use 
C(|iiation  (11)  provided  that  the  path  of  the  comet  is  not  nearly  in 
the  ecliptic.  AVhen  the  comet,  at  the  time  of  the  second  observation, 
is  in  quadrature  with  the  sun,  equation  (19)  becomes  indetenninatc 
in  form,  and  we  must  have  recourse  to  the  original  equation  (16), 
wliich  does  not  necessarily  fail  in  this  case. 

AVIien  both  equations  (11)  and  (IG)  are  simultaneously  nearly  in- 
ck'tcrniinate,  so  as  to  be  greatly  affected  by  errors  of  observation,  the 
relation  between  p  and  />"  must  be  determined  by  means  of  equation 
(15),  which  fails  only  when  the  motion  of  the  comet  in  longitude  is 
very  small.  It  will  rarely  happen  that  all  three  equations,  (14), 
(15),  and  (16),  are  inap})lic;ablc,  and  when  such  a  case  does  occur  it 
will  indicate  that  the  data  are  not  sufficient  for  the  determination  of 
the  elements  of  the  orbit.  In  general,  equation  (16)  or  (19)  is  to  be 
used  when  the  motion  of  the  comet  in  latitude  is  considerable,  and 
equation  (15)  when  the  motion  in  longitude  is  greater  than  in  latitude. 


174 


THf:ORETICAL   ASTRONOMY. 


(M.  The  fornuilif  iilroady  derived  are  suffioiont  to  determine  tlio 
reL'iion  Ix-tween  •>"  and  (>  wlien  tlie  values  of  n  and  n"  are  known, 
and  it  remains,  tlieretbre,  to  derive  the  expressions  lor  these  ijuan- 
tities. 

If  we  put 

k  m  ~t)  ^  t", 

k  (t"  —  t')  =  r,  (20) 

k  (t"  - 1)  =-  r', 

and  express  the  values  of  x,  y,  z,  x",  y",  z"  in  terms  of  a;',  t/',  z'  by 
expansion  into  series,  wo  liave 


a;  1=  a; 


ax'    r"    ,      1      f/V    r' 


,"S 


1  f/V      T 


'»~'     J'a 


(H  '  h  ^1.2'  dV  '  k'        1.2.3'  df 


h^     r    .     1     f/V     r' 


^'  +  -;/.-T  +  T 


1        d!>x'     T» 


(/«     yfc    '   1.2     dt'     L^     '   1.2.3     di^     /r* 


4-  &c., 


(21) 


and  simihir  expressions  for  ?/,  i/",  3,  and  z".  We  sliall.  however,  take 
the  phuie  of  the  orbit  as  the  fundamental  plane,  in  whieh  case  s,  ;', 
and  z"  vanish. 

The  fundamental  equations  for  the  motion  of  a  heaveidy  body 
relative  to  the  sun  are,  if  v.e  neglect  its  mass  in  comparison  with 
that  of  the  sun, 

dV      Pa/ 

df'  +  r"  ""    ' 

d'y'   ,  ^V 


df    '    r"' 


0. 


If  we  differentiate  the  first  of  these  equations;  we  get 


(Px'  _  3/tV   dy_  P   dx' 
df'  """  '7*'  '  dt       r"  '  ~dT 


Diilerentiating  ngain,  we  find 

rfV  __  / 1       12k'  I  dr^  \ »      3^^    d.V  \   ,  ,  6^ 
df  ~  \  r"  ""  " ?»'  \  "(It  f  ■+"  -/♦  ■  dt'  }  "^    ^  r'* 


6/fe»    di-;^    d3[ 
dt  '  dt' 


d'l/ 


Writing  y  instead  of  x,  we  shall  have  the  expressions  for  ■--:  and 

--—•     Substituting  these  values  of  the  differential  coefficients  in  eqiia- 

tions  (21),  and  the  eorres})onding  expressions  for  y  and  y",  nnd 
putting 


-^  1 


^11 


r-"i 


ORBIT   OF  A   HEAVENLY   BODY. 

-'"     dr' 


175 


•  r'*       ^  kr*  '  dt 


,"3 


t"«       (//•' 


.2  ^       ^/ 


(22) 


h"  -- 1  _ .  ^'  4- 1  i!-  'K 

^   '"  k       Hr'''^^kV'*'~dt ' 


wc  obtain 


X  --—  ax 


,  dx' 


ui'  =  a'V  -i-  b" 


dx' 
'fit' 


y  =  ay'~b 


dij 

di' 


f  =  ay-i-b"^. 


From  these  equations  we  easily  derive 

,           ,          ,  x'di/  —  y'dJ 
t/x  —x'y  =h — •'—^ , 

,    ,       ii.x'dy'  —  y'dx' 

^-^  —dr~' 


y"x' 


(23) 


fx  —  x"y  =  {ab"  +  a"b) 


///A  ^'<^y'  —  y'doi^ 


dt 


Tiic  first  members  of  these  equations  are  double  the  areas  of  tlie 
triaiitflos  formed  by  tlie  radii -vcctores  and  the  chords  of  the  orbit 
ootween  tlie  places  of  the  comet  or  planet.     Thus, 

yx-x'y^irr'l         y"x' -  x"y' ^Ir'r^'l         fx~a^'y^{rr"-\,     (24) 

and  x'dy'  —  y'dx'  is  double  the  area  described  by  the  radius-vector 

x'dv' n'dv' 

during  the  element  of  time  di,  and,  consequently,  - — - —    ''        ia 

double  the  areal  velocity.  Therefore  we  shall  have,  neglecting  the 
mass  of  the  body, 

x!d]f  —  ifdx' 


M 


V^ki/p, 


in  which  p  is  the  semi-parameter  of  the  orbit.     The  equations  (23), 
tlicrofore,  become 

['•*•']  ^  hk  ^p,  [rV]  =  b"k  y/p,  [r7-"]  =  {ab"  -{-  a"b)  k  y'p- 

Substituting  for  a,  6,  a",  b"  their  values  from  (22),  we  find,  since 

r'  -    -  -I-  t" 


176 


TlIKOllKTICA  Ti   A.STIIONOMY. 


['•'•"]  =r'v-(i-^;;!J+ 


r'      dr' 
/c7' '  dt  ' ' 
r'Hr  —  r")    dr' 


/cr'* 


dt 


...). 


(25) 


r  ■'  f"i  r  'j''i 

From  tlioso  o(| nations  tlio  values  of  h  =;  r- -77.1  'Hi^^  '»'" '^- T^'^n^  'n^^v 

[?•/•  J  Irr  ] 

be  (h'l'ivc'd  ;  and  tin;  results  arc 


(2Gj 


?i  = 


which  values  arc  exact  to  the  third  powers  of  the  time,  inclusive. 
In  tiic  case  of  the  orbit  of  the  earth,  the  term  of  the  third  o'M    . 

bein<j;  multi[)licd   by  the  very  fjmall  (juantity      .  >  is  reduced  t<>  „ 

superior  order,  and,  therefore,  it  may  be  uc<;lectcd,  so  that  in  tlii- 
casc  wc  shall  have,  to  the  same  degree  of  aj)proximation  as  in  (2()), 


^27) 


[rV] 


From  the  e(iuatious  (20)  or  from  (25),  since  -7,  ~-  f — ,--,-   we  find 
1  \     /  yj         L''''J 


n  ^  -  / 


+  1 


r'  -i-  r'"    dr' 
kr'*     '  'dt 


■■■)■ 


(28) 


dr' 


Since  this  equation  involves  /•'  and     .-.  it  is  evident  that  the  value 

of  -,,.  in  the  case  of  an  orbit   wholly  unknown,  can   be  deteni. 

only  by  successive  approximations.     In  the  first  approximation  "1 
the  elements  of  the  orbit  of  a  heaveidy  body,  the  intervals  bctwcin 
the  oi)servations  will  usually  be  small,  and  the  series  of  terms  of  i2>* 
will  converge  J'apidly,  so  tliat  we  may  take 

rt    _    T 


ORBIT  OF   A   HEAVENLY   BODY. 


177 


and  similarly 


N 
N" 


T 


Houce  tho  equiition  (11)  reduces  to 


P"  =  ^„IM'P. 


(29) 


It  will  be  ol)ser%-ed,  further,  that  if  the  intervals  botAvcen  the  observa- 
tions  are    equal,   the  tern     of  the   second    order    in   equation   (28) 

vanishes,  and  the  su])positinn  that  —,  =  —,  is  correct  to  terms  of  the 

third  order.  It  will  be  advantageous,  therefore,  to  select  observa- 
tions whose  intervals  approach  nearest  to  equality.  But  if  tho 
observations  available  do  not  admit  of  the  selection  of  those  which 
give  nearly  ecjual  intervals,  and  these  intervals  are  necessarily  very 
uno(iual,  it  will  be  more  accurate  to  assume 


n 

n" 


N_ 
N'" 


and  compute  the  values  of  N  and  N"  by  means  of  equations  (9), 
since,  according  to  (27)  and  (28),  if  r'  docs  not  differ  nuich  from  R\ 
till'  error  of  this  assumption  will  only  involve  terms  of  the  third 
order,  ev<'n  when  the  values  of  r  and  r"  differ  very  much. 

Whene\er  the  values  of  p  and  o"  can  be  found  when  tiiat  of  their 
rati'*  is  given,  we  may  at  once  derive  the  corresponding  values  of  r 
anil  r",  as  will  be  subsequently  ex])Iained. 

The  values  of  /•  and  /•"  may  also  be  expressed  in  terms  of  t'  by 
meuns  of  scries,  and  wi-  have 


dr' 


•iJ         -2 


from  which  we  derive 


/' 


neglecting  terms  of  the  third  order.     Therefore 

r'  ■      k  (r"  -  r") 


<// 


(30) 


12 


178 


THEORETICAL  ASTRONOMY. 


and  when  tlie  intervals  are  equal,  this  value  is  exact  to  terms  of  the 
fourth  order.     We  have,  also, 


which  gives 


i(,.  +  r")-AO-"-»-) 


(31) 


Therefore,  when  r  and  r"  have  been  determined  by  a  first  approxi- 
mation, the  approximate  values  of  r'  and  -.,-  are  obtained  from  these 
equations,  by  means  of  which  the  value  of  -;;  may  be  recomputed 

"it 

from  equation  (28).     We  also  compute 


N  _ R'E"shUQ"—Q') 
N''~   EI{'s\n(Q'—0)' 


(32) 


n 


N 


and  substitute  in  equation  (11)  the  values  of  -y,  and  ^^r,  thus  found. 

If  we  designate  by  31  the  ratio  of  the  curtate  distances  p  and  //', 
y:e  have 

^=^I'  =  ,,.»+,r(j-|,)f.  ,  (33, 

In  the  numerical  application  of  this,  the  approximate  value  of  />  will 
be  used  in  computing  the  last  term  of  the  second  member. 

In  the  case  of  the  determination  of  an  orbit  when  the  approximate 

elements  are  already  known,  the  value  of  -77  may  be  computed  from 

ih 


n 

"77 


n 


rr"sin(v"      (0 
rr'  sm{v'  —  v)  ' 


(34) 


iV 


and  that  of  ^-y,  from  (32);  and  the  value  of  31  derived  by  means  of 
these  from  (33)  will  not  retpiire  any  further  ci^rrection. 

65.  When  the  apparent  path  of  the  binly  ii^  ^*»K*h  that  the  value 
of  3r,  as  derived  from  tbe  first  of  etiualions  (10^  is  either  indeter- 
minate or  greatly  aftVeti-c  by  errors  of  observHtii«*i.  the  equations  (15) 
and  (16)  must  be  employj'u.  The  ]a^^t  temfc  f4"  t'lese  equation,  may 
be  changed  to  a  form  which  is  more  convtmienl  in  the  approximations 
to  the  value  of  the  ratio  of  jf/'  to  ft. 

Let   Y,  Y'j  Y"  be  the  ordinate^  ai  the  sun  when  the  Jixis  of 


ORBIT  OF   A   HEAVENLY   BODY. 


179 


ab-cissas  is  directed  to  that  point  in  the  ecliptic  whose  longitude  is 
/,',  and  wc  have 

Y  ^E  sin(0  —A'), 

Y'  ^B'  ainO'  —X'), 

Y"^E"Bm(,Q"  —  n. 

'Sow,  in  the  last  terra  of  equation  (15),  it  will  be  sufficient  to  put 


n 


n 

~>7 


N"' 


and,  introducing  Y,  Y',  1  ",  it  becomes 


( 


-^r  Y-lrY'+  F"]cosec(A"-A'). 

1 


(35) 


It  now  remains  to  find  the  value  of  -77-     From  the  second  of  equa- 


?/. 


lions  (26)  we  find,  to  terras  of  the  second  order  inclusive, 


We  have,  also, 
and  hence 


n 


N" 


Therefore,  the  expression  (35)  becoraes 

But,  according  to  equations  (5), 

NY—Y'+N"Y"^0, 
and  the  foregoing  expression  reduces  to 

.  1 1!lrr'4-r"^/ 1 1  \J?'sin(Q^-;/) 

"*■  ^^  t"  ^    "•"  ^  \  r'»       E"  j     sin  (A"  -  X')    ' 

since  Y'  —  B'  sin(0'  —  A').     Hence  the  equation  (15)  becomes 

„_     n      sin  (X'  - ;,)        ,  rr^  /I         1  \  jg^  sin  (A^-QQ 

'^  ~  '^  n"  ■  sin  (A"  -  A')       5 "?'"  ^^"^  "T"  "^  ^  \  73      ii'3  /  ^  sin~(A"  -  A')   "    ^-  "^ 


180 

If  wo  put 


TIIEORETICAI.   ASTRONOMY. 


,r 11      sin  (A'  —  k) 


1,"     rz- 


^.^l_.:L_.i;,(.'+,'0!^^f^'-O')   R'll       1 


11 


sin  (A'  —  A)       /^ 


\7^      jK'O' 


we  have 


(37) 


Lot  us  now  oonsidor  tlio  equation  (IG),  and  lot  us  dosignato  by  A', 
X',  X"  the  ahs(!issas  of  th(!  earth,  the  axis  of  abscissas  l)oinfj;  direotod 
to  tliat  point  of  the  ecliptie  tor  which  the  lojigitude  is  ©',  then 

X  ==Rcoii(Q-Q'), 
X'  =  M', 

X"=R"cos{Q"-Q'). 

It  will  be  .sufficient,  in  the  last  term  of  (IG),  to  put 


n 

"77 


n 


N 


1    . 


and  for  -,-,  its  value  in  terms  of  N"  as  already  found.     Then,  since 

NX-X'-^N"X"=0, 


R'tanfi' 


this  term  reduces  to 

_  .  -'  (>'  ,    ,"  (  1  _  ±  \ —  _- 

8  7"  ^     ^  "^  ^  \  73       ie'3 ;  tan  /'/'  COS  (A'—  ©')  -  tan  fi'  cos  (A"—  ©')  ' 

and  if  wo  put 


Af>  —  JL      t«"  P'  cos  (^  —  Q' )  -^tiui  /? co3(A'— Q') 
■*'»  ~  n"' '  till)  /J"  cos  (A'  -  O')  —  tiui  ii'  cos  (A"  —  O')' 


tan  ft' 


(38) 
22' 


'   "'      *   71      T"^"^     Hr"       ii''/tan/3'co8(;i-0')— tan/'ico8(;i'— O')     p' 
the  equation  (16)  becomes 

(39) 


M=='-^Af'F'. 


In  the  numerical  application  of  these  formulaj,  if  the  elemeiits  are 
not  api)roxinnitely  known,  we  first  assume 

n  T 


when  the  intervals  are  nearly  equal,  and 


ORBIT  QF  A   HEAVENLY    BODY. 


181 


n 


N 


as  given  by  (32),  wliou  tlic  iiitorvals  are  very  iinef(iial,  and  ncj^lccfc 
IIk^  llictors /'' and  F'.  TIk;  values  of  ,o  and  |r>"  wliicli  are  thus  ol)- 
tained,  enable  us  to  find  an  ai)j)roxiniate  value  of  /•',  and  with  this  a 

more  exaet  value  of-.-,  may  be  found,  and  also  the  value  of  F  ov  F'. 

Whenever  equation  (11)  is  not  materially  atfeeted  by  errors  of 
observation,  it  will  furnish  the  value  of  i)[  with  more  aeeuraey  than 
the  e<[uations  (37)  and  (39),  sinee  the  negleetcid  terms  will  not  be  so 
great  as  in  the  ease  of  thess  e(juations.  In  general,  therefore,  it  is  to 
he  prefeiTcd,  and,  in  the  ease  in  whi(!h  it  fails,  the  very  eireumstance 
tliiit  the  geocentrie  j>atli  of  the  body  is  nearly  in  a  great  eirele,  makes 
tlie  values  of  i<' and  F'  diller  but  little  from  unity,  sinee,  in  order 
tl.'it  the  apjiarcnt  path  of  the  boily  may  be  ncjarly  in  a  great  eirele, 
'/•'  must  differ  very  little  from  W. 

66.  AVheu  the  value  of  3£  has  been  found,  we  may  proeeed  to 
determine,  by  means  of  other  relations  between  (t  and  ft",  the  values 
of  the  (|uantities  themselves. 

The  eo-ordinates  of  the  first  place  of  the  earth  referred  to  the  third, 
are 

2/,  =  i«;"sinO"  — /isinO.  ^     ^ 

If  we  represent  by  g  the  chord  of  the  (uirth'.-i  orbit  between  the  places 
corresponding  to  the  first  and  third  observations,  and  l)y  (r  the  longi- 
tude of  tlie  first  place  of  the  earth  'as  seen  from  the  third,  we  shall 
have 

x,  =  g  cos  G,  yi=^  9  sin  G, 

and,  consequently, 

R"  cos  CO"  —  O)  —  i?  =  .7  cos  ( G    -  O),  .... 

ii"sin(0"-0)  =^8in(6!-0).  ^     ^ 

If  "]/  represents  the  angle  at  the  earth  between  the  sun  and  comet 
at  the  first  observation,  and  if  we  designate  by  w  the  inclination  to 
the  ecliptic  of  a  plane  passing  through  \\\v,  places  of  the  earth,  sun, 
and  comet  or  iilanet  for  the  first  observation,  the  longitude  of  the 
ascending  nodt  of  this  plane  on  the  eclij)tic  will  be  O,  and  we  siiall 
have,  in  accordance  with  e(]uations  (81),, 

cos  4  n=:  COS  /9  cos  (A  —  ©  ), 

sin  4  cos  IV  =  cos  /?  sin  (A  —  O), 
sin  ■i  sin  w  =;  sin  ,5, 


182 

from  wliich 


TII EORETICAL   ASTIIONOM Y. 


tan  IV  = 
tau4  = 


_tan  i? 

sTnCA--©)' 
tana  — O) 

cos  IV 


(42) 


Since  008/9  is  always  jmsitivo,  cos  4-  and  cos(,'> —  ©)  must  have  the 
same  sif^n;  and,  further,  -v^  cannot  exceed  180°. 

In  the  same  manner,  if  «/'  and  ■\l/"  rei)resent  analogous  quantities 
i'or  the  time  of  the  third  observation,  we  obtain 


tan  w"  = 


tan/S" 


sin  (A"  -  ©")' 
tan  (k"  —  0") 


Wc  also  have 


tanV'  = 

cos?o' 

COS  4"  ==  cos  /5"  cos  (r  —  O"). 
r'  =J^-{-E'—  2JE  cos  ^, 


(43) 


which  may  be  transformed  into 

r'  =  (p  sec  13  — li  cos  ^y  +  iiJ^  siu»  4 ; 
and  in  a  similar  manner  we  find 

r"'  =  (p"  sec  /5"  —  li"  cos  +")» +  li'"  sin=  4". 


(44) 


(45) 


Let  n  designate  the  chord  of  the  orbit  of  the  body  between  the  first 
and  third  places,  and  we  have 


But 


X'  -  {^"  -  ^y + (y"  -  yy + iz"  -  zy 


«  =  />  cos  A  —  J?  cos  O, 
y  ^=  P  sin  A  —  i2  sin  ©, 
g  z=  /)  tan  /S, 


and,  since  ()"  =  Mo, 


x"=  Mp  cos k"—R"  cos  ©", 
2/"  =  iW"/>  sin  A"  —  i?"  sin  ©", 
s"  =  il//>tan/5" 

from  which  we  derive,  introducing  g  and  (r, 

x"  —  x  =  Mp  cos  A"  —  /)  cos  A  — ■  _r/  cos  G, 
y"  —  y  =  3//)  sin  A"  —  |r>  sin  A  —  rj  sin  (?, 
2"  —  2  =  Mp  tan/5"—  p  tan/?. 


Let  us  now  put 


Tlien  we  have 


ORBIT  OP  A    HKAVKNLY    MODY. 

Mf>  cos  X"  —  /;  cos  ^      v:  /)Ji   C'OS  X  COS  ff, 

Ml>  sill  >■"  —  p  sill  k  ^.  fill  cos  C  sin  i/, 
Jl//*  tan  fi" —  p  tan  ,3  =  /^/j  sin  C. 

af'  —  a;  =^  /)/i  cos  C  cos  /T"  —  j/  cos  G, 
i/'  —  y  --  fih  cos  ;  sin  H —  y  sin  fr, 

2"  —  2  -^  /'/i  sin  JT. 


183 

(40) 


Squaring  tlicse  values,  and  adding,  we  get,  by  reduction, 
x'  =  ^»/i'  —  2ff  ph  cos  ?  cos  (  (?  —  if)  -|-  y'^ ; 


and  if  we  put 
wc  have 


cos  C  cos  ( G  —  //^  =  cos  9", 
x'  =:;  (/>/t  —  g  cos  ^)'  +  (jr*  sin'  <p. 


(47) 

(48) 
(49) 


If  we  multiply  the  first  of  equations  (4G)  by  coh?/',  and  tlie 
second  by  sin  A",  and  add  the  products;  then  multiply  the  first  by 
sin/",  and  the  second  by  cos^.",  and  subtract,  we  obtain 


h  cos  C  cos  (H  — ;.")  =--  M  —  cos  (A"  —  A), 

h  cos  C  sin  (//—  X")  =  sin  ( /"  —  A), 

/t  sin  C  =  M  tan  /3"  —  tan  /?, 

by  means  of  which  we  may  determine  /(,  ^,  and  II. 
Let  m?  now  put 


(50) 


g  cos  <f 


g  sin  ^  =  J., 

/t  cos  ,9  =  />, 
A_co.^ 
M 

hR  cos  4-  =  c,  jf  cos  ^  —  b"R"  cos  4."  :=  c", 

ph  —  g  cos  ^J*  =  (/, 


R  sin  4  =  -B, 
i?"sinV'=-5", 


ft", 


(51) 


and  the  equations  (44),  (45),  and  (49)  become 


(52) 


i? 


The  equations  thus  derived  are  independent  of  the  form  of  the 
orbit,  and  are  applicable  to  the  case  of  any  heavenly  body  revolving 
around  the  sun.  Thev  will  serve  to  determine  r  and  /•"  in  all  eases 
in  which  the  unknown  quantity  d  eau  be  determined.   If />  is  known. 


184 


TIIKOIJKTICAIi    ASTKONOMY. 


(/  hccomos  known  directly;  l)nt  in  tlu;  case  of  an  unknown  orhit, 
tlicsc  c'(|natioii.s  arc  a|)|tli(al)lc  only  when  //  or  >l  may  l)c  determined 
directly  or  indirectly  I'ntm  tlie  data  Inrnished  l)y  observation. 

G7.  Since  (lie  e(|iiation.s  (512)  involve  (w(t  radii-vectores /•  and  /•" 
and  tli<'  chord  K  joining;  their  extremities,  it  is  evident  that  an  addi- 
tional ei|nation  involving-  these  and  known  <|Uantities  will  enahle  uh 
to  derive  d,  it"  not  <lireetly,  at  least  hy  su<'eessive  approximations. 
There  is,  indeed,  a  remarkahle  relation  existinff  Ix'tween  two  radii- 
vectoros,  the  chord  joinin}f  their  extnunities,  and  the  time  of'chwcrihin^ 
the  part  of  the  orhit  included  by  these  radii-veotores.  In  jjjcneral, 
the  e(piation  which  expresses  this  relation  involves  also  the  semi- 
transverse  axis  ol'  the  orbit ;  and  hence,  i,i  the  ease  of  an  unknown 
orbit,  it  wmII  not  be  sullicient,  in  connection  with  the  eipiations  (02), 
for  the  determination  of  <l,  unless  sonu;  assumption  is  made  in  rcfrard 
to  the  value  of  the  senn-transverse  axis.  For  the  sj)ecial  ease  of 
parabolic  motion,  the  semi-transverse  axis  is  iidiintc^,  and  the  resnlt- 
injx  eipiation  invttlves  only  the  time,  tlu^  two  radii-vcetores,  and  the 
chord  of  the  part  of  tlu;  orbit  included  by  these.  It  is,  therefore, 
ada[)ted  to  the  deternunation  of  the  elements  when  the  orbit  is  sup- 
posed to  be  a  parabola,  and,  though  it  is  transcendental  in  form,  it 
may  be  easily  solved  by  trial.  To  determine  this  expression,  let  us 
resume  the  equations 

=  tan  ji'  +  i  tan'  hv 


and,  for  the  time  t", 

k{t"—T) 
V2(^ 


=  tan  y  +  ^  tan'  U'. 


Subtracting  the  former  from  the  latter,  and  reducing,  wo  obtain 

V2q'^-  cos  oc"  cos  vjc  \  7        cos  . 1 1;"  cos  If       qj' 


and,  since  r  =  q  sec'-^r,  this  gives 


lk(t"  —  f)       smHv"—v)\/: 


—  (  r  -f  r"+  cos  I  (v"—v)Vrr"Y  (53) 


1/2  Vq 

But  we  liavc,  also,  from  the  triangle  formed  by  the  choi'd  x  and  the 
radii-vectores  r  and  r", 

j,2  ^  J.2  _j_  r"-i  _  2rr"  cos  (v"—  v) 
:=  (r  -f  r"y  —  4rr"  cos'  \  (v"  —  v). 


PAIlAnOLIC  OHIIIT. 


185 


Tlicretbrc, 


cos  A  (v  —  v)  =  -±    — ! ....  -^ . 

Let  118  now  put 

r  '\-  r"  +  X  :_.  )u\  r  +  r"  —  x  =  »i', 

w  iiiid  n  being  positive  (pmntitier*.     Tlion  we  sliull  Iiavo 

2  cos  .1  ( v"  —  V)  V^rV  :=:--  lI  ■  »ui ; 


(r)4) 


and,  since  m  and  n  :irc  alwiiys  positive,  it  follows  that  the  upper  sij^u 
iiiiist  1)0  used  when  r" —  v  is  less  than  ISO",  and  the  lower  sij^u  when 
v"  —  r  is  ifreater  than  180^.  Combining  the  last  eijuation  with  (-Vj), 
llio  result  is 

.5a;  U  —  0  =  — ;— (m*  +  »i^  ±  mil).  (oo) 

T/2<^ 

Now  wc  have 

sin  h  (y" —  v)  =  sin  Id"  cos  ^v  —  cos  !^v"  sin  U: 
8([uariiig  this,  and  reducing,  we  get 

sin'  \  (v"  —  v)  =  cos"  iv  +  cos'  Iv"  —  2  cos  {v"  cos  Jv  cos  ^  (o" — v), 
or,  introducing  r  and  (/, 


JttM 


Therefore, 


sin'  ,Ui. '  —  r)  =  i  +  -4,  -h  «7  ■  /;• 
>•       /'  rr 


sin  ;V  (f  —  v)  =       ._!.  -  (w  -t-  w). 
2/y/' 


Introducing  this  value  into  equation  (55),  we  find 

mt"  ~  t)  ^  m^  :^  n\ 

Replacing  m  and  n  by  tlieir  values  expressed  in  terms  of  ?•,  r",  and 
X,  this  becomes 


Qkit" -t)  =  (r  +  r"  +  x)t  =p  (r  +  r" -  x)J, 


(56) 


the  upper  nifa  being  used  when  v"  —  v  is  less  than  180°.  This 
equation  'xprcises  the  I'elation  between  the  time  of  describing  any 
parabolii!  arj  and  the  rectilinear  distances  of  its  extremities  from  each 
other  and  from  the  sun,  and  enables  us  at  once,  when  three  of  these 
quantities  are  given,  to  find  the  fourth,  independent  of  either  the 


IMAGE  EVALUATION 
TEST  TARGET  (MT-3) 


1.0 


I.I 


IS  ■"   1^ 
■"  Ui    |2.2 

12.0 


t   Ifi 


IL25  i  1.4 


141 
1.6 


<v^ 


7 


'/ 


Hiotographic 

Scieices 

Corporation 


23  WIST  MAIN  STRUT 

WIBSTIR.N.Y.  MStO 

(716)  •73-4S03 


w 

^ 


^ 

.€^ 

^"v^ 

^"^% 


^ 


^ 


6^ 


^ 


186 


THEORETICAL   ASTRONOMY. 


jM'rilu'lion  distance  or  the  position  of  the  perihelion  with  respect  to 
the  arc  doscril)e(l. 

68.  The  tninscendental  form  of  the  equation  (56)  indicates  that, 
when  cither  of  the  quantities  in  the  second  member  is  to  he  fuiuid, 
it  must  be  solved  by  successive  trials;  and,  to  facilitate  these  approxi- 
mations, it  may  be  tmnsfbrmed  as  follows: — 

Since  the  chord  x  can  never  exceed  r  +  /•",  we  may  put 


r +  / 


>-  =Binr. 


f57) 


and,  since  x,  r,  and  r"  arc  jwsitive,  sin;-'  must  always  be  positive. 
The  value  of  ;•'  must,  therefore,  be  within  the  limits  0"  and  180°. 
From  the  last  equation  we  obtain 


(r  +  r"r  -  x» 


and  substituting  for  k'  its  value  given  by 


this  becomes 


x»  =  (r  +  r"y  —  4rr"  cos'  {  {v"  —  v), 


=>«' 


cos'  Y  = 


4rr"  cos»  '.(i/'  —  v) 


Therefore,  we  have 


{r^r") 


fi\i 


COS 


and  also 


tan  /  = 


2Vrr"coa\{v"  —  v) 


(58) 

(59) 


Hence  it  appears  that  when  v" — v  is  less  than  180°,  y'  belongs  to 
the  first  quadrant,  and  that  when  v" —  v  is  greater  than  180°,  cos;*' 
is  negative,  and  y'  belongs  to  the  second  quadrant. 

If  we  introduce  /-'  into  the  expressions  for  vi^  and  n^,  they  become 


which  give 


m»=(r-f  r")(l  +sin/), 
n'  =  (r  +  r"Ml— sinA 

m*=(r-\-  r")  (cos  ^  +  sin  .y)», 
n'=ir-\-  r")  (±  cos  hj'  ^  sin  i, ')»; 


and,  since  y'  is  greater   than  90°  when  v"  —  v  exceeds  180°,  the 
equation  (56)  becones 


6/ 


(r  +  r")* 


=  (cos  \r'  +  sin  {y'f —  (cos  \y'  —  sin  yf. 


PARABOLIC  ORBIT. 

From  this  ccjuation  we  got 
6t' 


187 


or 


(r  +  r")^ 
6/ 


6  cos'  A/  sin  \y'  +  2  sin'  \y', 
=  6  sin  \/  —  4  sin'  1/ ; 


(r  4-  r")l 
and  this,  again,  may  bo  transformed  into 

2l(,.  +  r")^^         \    ]/2   /         \    1/2    / 
Let  US  now  put 


or 

autl  wc  have 


3r' 


V^2(r  +  r")^ 


sin  V 
8ma;  =  —  --, 

V  2 


sin  y  =  V2  sin  a;, 


=  3  sin  a;  —  4  sin'  a;  =^  sin  3a;. 


(60) 


(61) 


(62) 


When  v"  —  V  is  less  than  180°,  y'  must  be  less  than  90°,  and 
'lenco,  in  this  case,  sin  x  cannot  exceed  the  value  J,  or  x  must  be 
within  the  limits  0°  and  30°.  When  v"  —  v  is  greater  than  180°, 
tlio  angle  y'  is  within  the  limits  90°  and  180°,  and  corresponding  to 
these  limits,  the  values  of  sin  .c  are,  respectively,  \  and  \\  2>  Hence, 
in  the  case  that  v"  —  v  exceeds  180°,  it  follows  that  x  must  be  within 
the  limits  .30°  und  45°. 

The  equation 

:  sin  3a; 


V2{i'  +  r")' 

is  satisfied  by  the  values  3.r  and  180°  --  3.r*;  but  when  the  first  gives 
.r  less  than  15°,  there  can  be  but  one  solution,  the  value  180°  — 3.i' 
being  in  this  case  excluded  by  the  condition  that  3.c  cannot  ex<'eed 
135°.  When  x  is  greater  than  15°,  the  required  condition  will  be 
Batisficd  by  3a;  or  by  180°  —  3a;,  and  there  will  be  two  solutiv.ns, 
eorresponding  respectively  to  the  cases  in  which  v"  —  v  is  less  than 
180'',  and  in  which  v"  —  v  is  greater  than  180°.  Consequently, 
wlion  it  is  not  known  whether  the  heliocentric  motion  during  the 
intervals"  —  <  is  greater  or  less  than  180°,  and  we  find  3x  grcjiter 
than  45°,  the  same  data  will  be  ssitisfied  by  these  two  diH'crent 
solutions.     In  practice,  however,  it  is  readily  known  which  of  the 


188 


THEORETICAL   ASTKONOMY. 


two  solutions  must  bo  adoptod,  sinco,  wlioii  tlio  interval  t"  —  t  in  not 
very   larj^c,  the  lielitMrntrio  motion  camiot  oxctnid   l.SO°,  nnlt-ss  tlu' 
])crili(.'lion  distanre  i.s  very  snuill ;  and  tlu;  known  circnmstancoti  will 
gcni'i'iklly  show  whet  her  sueh  an  asMimption  is  a(hnissil)le. 
Wu  shall  now  put 


2r^ 


aud  wc  obtain 

We  have,  also, 
and  hence 
Therefore 


(r  -I-  r") » 

8U1  <iX  —        -. 

sin  \/  =■  |/2''^ina^, 


(63) 

((J4) 


cos  \/  =  1/1  —  2  sin'  J*  =  l/co8  2x. 


Bin  /  =  2^  sin  x  V  cos  2x, 
and,  since  x  =  (?•  +  r")  sin ;-',  we  have 

H  =  2J  (r  +  r")  sin  x  v/cos  2x. 

3sinx    / — -_- 
M  =  - .    .,    K  cos  2x, 


If  wc  put 


(65) 


the  preceding  equation  reduces  to 


It' 


.^^^r  II. 


V  (>•  -f-  r") 


(66) 


From  equation  (64)  it  appears  that  ;y  must  be  within  the  limits  0 
and  J)  ^.  We  may,  therelbre,  construct  a  table  which,  with  r^  as 
the  arj^ument,  will  give  the  corresponding  value  of  //,  since,  witli  a 
given  value  of  jy,  3a;  may  be  derived  from  equati(»n  (<)4),  and  tlicii 
the  value  of  ft  from  ((55).  Tabic  XT.  gives  the  values  of  //  corro- 
eponding  to  values  of  jy  from  0.0  to  0.9. 

69.  In  determining  an  orbit  wholly  unknown,  it  will  be  necessary 
to  make  some  assiiniption  in  regard  to  the  approximate  distance  <»!" 
the  comet  from  the  sun.  *n  this  ease  the  interval  t"  —  I  will  gent- 
rally  Im)  small,  and,  conseq.\ently,  x  will  be  small  compared  with  r 
and  /'".  As  a  first  assumption  we  may  take /•  =- 1,  or  r -|- ;•"  —  2, 
aud  n  =  1,  and  then  find  x  from  the  formula 

X  =  tV2- 


PA  II A  no  I,  ir  OR  HIT. 


189 


With  this  value  of  x  we  eonipute  il,  r,  nnd  r"  by  means  of  the 
o(|Ua(i(ms  (o2).  iliiviiij;  thus  found  approximate  vahies  of  r  and  r", 
we  compute  jy  l)y  nu'ans  of  («l;J),  and  with  this  value  we  enter  Tal)lo 
XI.  and  tak(;  out  the  eorrespondinj;  vulue  of /i.  A  seeond  value 
fur  X  is  then  found  from  (()(>),  witli  which  we  recompute  rand  /•",  a'>d 
jirocced  as  iM'fore,  until  tiie  values  of  these  ((uantities  remain  un- 
(■li;in<fed  T!<c  final  values  will  exactly  satisly  the  ecpiation  (oO), 
iinil  wil.  enable  us  to  eoniplete  the  determination  of  the  orbit. 

Alter  three  trials  the  value  of  /•  +  '•"  niay  be  found  very  nearly 
I'drnrt  from  the  numl)ers  already  derived.  Thus,  let  //  Im;  the  true 
value  of  lo^  (r  +  r"),  and  let  aj/  Ihj  the  ditl'erenee  between  any 
a.».-iiined  or  approximate  value  of  y  and  the  true  value,  or 

Tlioii  if  we  denote  by  i/^'  the  value  which  results  by  direct  caleulatiou 
I'roiii  the  n8.sume<l  value  y„,  we  shall  have 

yo'  —  2/»  -^/Cy,.)  =/(y  +  ^y). 

Kxpaiuling  this  function,  we  have 

y«'  -  y.  --/(y)  +  ^  Ay  +  u  Ay»  +  &c. 

Hut,  since  the  equations  (o2)  and  ((><))  will  be  exactly  satisfial  when 
the  true  value  of  y  is  u.-4ed,  it  follows  that 

/(y)  =  o, 

and  hence,  when  :^y  is  very  small,  so  that  we  may  neglect  terms  of 
tile  .second  order,  we  shall  have 

y<^  —  y„  ==  ^  Ay  =  vl  (y„  —  y). 

Lot  lis  now  denote  three  .successive  approximate  values  of  log  (r  +  »'") 


yo  —  yo  = «. 

tlien  we  shall  have 


yo"  -  yo' 


»'. 


o  =  i4  fy,  —  y), 

o'  =  yl(y;-y). 

Eliminating  A  from  these  (Kjuatious,  we  get 

y  (a'  —  o)  =  a'y^  —  ay^, 


from  which 


y=y.-a'---=y„ 


a' 


o — a 


(67) 


190 


THKOItETKAI,   ASTIIOXOMY. 


Unless  the  nswiimcd  vnliirs  nrc  (•on.Bi<lcnil)ly  in  orror,  tlio  valiio  (if 
y  (tr  of  l<»;f  (/•  i  /•")  lliiis  found  will  Ih'  siinicifiitly  exact ;  l)iit  should 
it  l)e  still  in  error,  we  nisiy,  from  the  three  viilnes  which  a|>|iro\ini!ito 
nearest  to  the  truth,  derive  ;/  with  still  greater  aeeumcy.  In  tiie 
nunu-rical  a|»|»li«"jUion  of  this  eijuation,  a  and  a'  may  Ik?  expressed  in 
units  o"  the  last  «ieciiiial  place  of  the  lopirithms  employed. 

The  solul  "on  of  ecpiation  (•'it}),  to  find  t"  t  when  x  is  known,  i.s 
readily  etlw  ted  by  means  of  Table  VIII.     Thus  we  have 


3r' 


V^2(r-\-r")i 
and,  wlicn  y'  is  less  than  90°,  if  wc  put 


=:  gin  ;}r, 


we  get 

or 


-r      fin  3j! 

iV=    .      ,, 
sui;' 


r'=:\V2Xmir'^r-\-r")u 

\Vh('n  ;-'  excecd.s  90°,  we  put 

N'  =  sin  3a;, 

t'  =.  I  \/~2  N'  (r  +  r")h 


(68) 


and  we  have 


(69) 


in  wliieh  log  J]  2  9.()733937.  With  the  argument  y'  we  tako 
from  Table  VJII.  the  corresponding  value  of  A"  or  N',  and  liy 
means  of  these  equations  r'       k  (t"  —  t)  is  at  once  derived. 

The  inverse  problem,  in  which  r'  is  known  and  x  is  recpiirod,  nuiy 
also  be  solved  by  means  of  the  sjuue  table.  Thus,  we  may  lor  a  liivt 
approximation  put 

x  =  ^  l/2. 

and  witli  this  value  of  x  compute  <J,  r,  and  r".  The  value  of  ;-'  is 
then  found  from 


smy  = 


r-\-r"' 


and  the  table  gives  tlie  corresponding  value  of  iVor  iV.     A  second 
approximation  to  x  will  be  given  by  the  ecpmtion 


K!> 


K  = 


v/2'iVl/r-f>' 


PAR  Vnoiwf  ORBIT. 


191 


or  by 


3 
in  wliicli   loe— ,- 
V  2 


_   3  T'tiinf 


0.326G()()3.     Then  wc  rooompiito  tf,  r,  and  »•", 

niid  |)n»c«'<'<l  as  Iwforo  until  X  roinains  uncliaiij^od.     The  a|)|)n>.\inia- 
liims  aiT  liU'ilifated  hy  means  of  ('(jnation  ((17). 
It  will  l>e  obscrvttl  that  d  is  ('oinputotl  from 

an<l  it  should  be  known  whether  the  positive  or  negative  sign  must 
be  used.     It  is  evident  from  the  eijuation 

d  =  ph  —  7  cos  <p, 

siiipo  o,  h,  and  //  are  positive  quantities,  that  so  hmg  as  ip  (whieh 
must  1m'  within  the  limits  0°  and  180°)  excewls  90°,  the  value  of  d 
must  be  positive;  and  therefore  if  must  be  loss  than  90°,  and  ffvonip 
greater  than  />/*,  in  ord(?r  that  d  may  be  negative.  The  eiiuution  (47) 
shows  tli.;t  when  x  is  greater  than  y,  we  have 

g  cos  tp  <  yh, 

and  hence  d  must  in  this  case  be  positive.  But  when  x  is  less  than 
.7,  either  the  positive  or  the  negative  value  of  d  will  answer  to  the 
given  value  of  f.  and  the  sign  to  bo  adopted  must  be  determined 
ironi  the  physical  conditions  of  the  problem. 

If  W(!  sup|K)so  the  chords  r/  and  x  to  be  proportional  to  the  linear 
voloeities  of  the  earth  and  comet  at  the  nnddle  observation,  we  have, 
the  occcntricity  of  the  earth's  orbit  being  neglected, 


=w?. 


whieh  shows  that  x  is  greater  than  r/,  and  that  d  is  positive,  so  long 
«  r'  is  less  than  2.  The  comets  are  rarely  visible  at  a  distance  from 
.lie  earth  which  much  exceeds  the  distance  of  the  earth  from  the  sun, 
and  a  comet  whose  nvdius-vcctor  is  2  must  be  nearly  in  opj-.osition  in 
order  to  satisfy  this  condition  of  visibility.  Hence  cases  will  rarely 
occur  in  which  d  can  be  negative,  and  for  those  which  do  occur  it 
will  sicnorally  bo  easy  to  determine  which  sign  is  to  be  used.  How- 
ever, if  d.  is  very  small,  it  may  be  impossible  to  decide  which  of  the 
two  solutions  is  correct  without  comparing  the  resulting  elements 
with  other  and  more  distant  observations. 


192 


TIIKOIIKTICAL    ASTIIOXOMY. 


70.  WluMi  the  vnlvK's  of  /•  and  /•"  have  Ik'cm  fiiinlly  (lotormiiicd,  as 
jui-t  cxplaiiiocl,  the  exact  value  ot"  tl  may  Ix;  computed,  uiul  then  we 
have 


P  = 


d  -f-  ff  C08  y» 


p"=Mp, 

from  whicli  to  find  ft  and  f»". 

According  to  the  tujuatioiirt  (90),,  wc  have 

r  009  b  C09  (/  —  0)  ::^-  />  C08  ( >  —  O)  - 

r  c().s6  sin  (/  —  O  )   -  /'  sin  (^  —  0)i 
}•  8in  b  =^  (>  tan  ,i. 


(70) 


R, 


and  al 


80 


r"  coHi"  co9(r-  O")  =  p"  co8{r—  O")  -  R', 
r"  cos  //'  sin  {I"  -  O")  -  //'  sin  (A"  -  Q"), 
r"Hin6"  --//'tan,3r", 


(71. 


(72) 


in  wliich  /  and  /"  are  the  lieliocontric  longitudcfl  and  b,  h"  the  corie- 
sponding  hcliwentric  hititudcs  of  the  comet.  From  these  c(jnati((iis 
we  find  /•,  /•",  /,  /",  b,  and  b"  -,  and  the  vahies  of  r  and  r"  thuH  found, 
should  agree  with  the  final  values  already  ol)tained.  When  I"  is  less 
than  /.  the  motion  of  the  comet  is  retrognule,  or,  rather,  when  the 
motion  is  such  that  the  heliocentric  longitude  is  diminishl  ig  instead 
of  increasing. 

From  the  tHpiations  (82),,  we  have 


tan  i  sin  (/  —  Ji)  =  tan  b. 
tan  i  sin  (/"—  ft )  =  tan  6", 


(73) 


which  may  lie  written 


■±.  tan  J  (sin  (/  — a;)co8(a;  —  ft  )  "|-  ^^in  (^*  —  JJ  ^  cos(/  —  a;))  =  tan6, 
±  tan  / (sin  ( t'—  x)  cos (x  —  ft )  -|-  sin  («  —  ft )  cos  {f—  x))  —  tan  b". 

Multiplying  the  first  of  these  equations  by  sin(^" —  .r),  and  the  second 
by  —  sin  (/  —  x),  and  adding  the  products,  we  get 

■±.  tan i  sin (x  —  ft )  sin {H'  —  /)  =  tan  b  sin (/"  —  x)  —  tan b"  sin (l  —  x); 

and  in  a  similar  manner  wc  find 

±  tan  i  cos  («  —  ft  )  sin  (f '  —  0  =  tan  b"  cos  {I  —  a;)  —  tan  6  cos  {I"  —  x). 

Now,  since  x  is  entirely  arbitrary,  we  may  put  it  equal  to  /,  and  we 
have 


l.U:  ORHIT.       . 

103 

tnii  h, 

tun  6"—  Uinb  vtmil"  ~ 

I) 

(74; 

niuir-t) 

1 

tan  J  sill  it —  Q)  =  ± 
tan  /  cos (/  —  Q)  =--  ± 


the  lower  sijxii  Ix-in^  ust'tl  when  it  i.s  doiiirt'd  to  introduce  tlu;  distinc- 
tion ol"  retrograde  motion. 

Tlie  torinnlie  will  lu'  better  adapted  to  io^aritliniie  ealeidation  it" 
wo  put  X  -  J(/"+  /),  whence  I"-  x  ^(1"  -  I)  and  /  -.r  \[l  -^  /"); 
ant!  we  obtain 

.  .    ,,,,„,,        ^v  ^i»  ^tt"  '[•  b) 

^  •  '  2  e«)M  ft  cos  ft    eo.x  \(i  —  /  )        ^ . 

^.    /•»     .       IX  ^1  «"•  »ft      ft) 

tan*  cosl.',  (/  +  O  —  Jj)  =  ±  o   — ,         ,„   .    .  ,  „, jx- 

^'        '  '  2  cos  ft  cos  ft   sui  \  ( r  —  /) 

Tliese  ocpiations*  may  also  he  derived  directly  i'rom  (I'.l)  hy  addition 
and  sui)tracti()ii.     Tims  we  have 

±  tan  /(sin  (/"—  ft  )  +  j*in  (/  —  ft  ))  =^  tan  ft"  +  tun  ft, 
±  tun/ (sin  (r—  ft)  —  8ia(/—  ft))  =  tan  ft"— tan  ft; 

and,  fince 

nn  (/"—  ft)  +  si" (/  -  ft )  --  2  sin  A  (r+  /  -  2ft )  cos  \  (l"~  I), 
HinCr-  ft )  —  8in(/  -  ft)  ^-=  2  cos  .J  (/"+/-  2ft )  sin  \  (l"-~  I), 


these  become 


*      •   •    /!/;"  I    /^       r.\       ^  Utan  ft" -f  tan  ft) 
tant8m(i(/'  +  0-ft)  =  ±      ,.„j,  ,(/"_/)-   • 


tan  /  cos  (J  (/"  +  /)— ft)  =--± 


^(  tin)  ft"-  tan  ft) 
"singer— 0     ' 


(76) 


which  may  be  readily  transformed  into  (75).  However,  since  h  and 
It''  will  be  found  by  means  of  their  tanji-ent.s  in  the  numerical  appli- 
cation of  ecpiations  (71)  and  (72),  if  addition  and  sulitraetion  loo;a- 
ritlunx  are  used,  the  e([uations  last  derived  will  l)e  more  convenient 
than  in  the  form  (75). 

As  soon  as  ft  and  i  have  been  computed  from  the  prccodini?  c(|ua- 
tions,  we  have,  for  the  determination  of  the  arguments  of  the  latituile 
n  and  Ji", 


tan  u  =  ±: 
Now  wc  have 


tana-ft)^  tann"=d:^^-'"-«J.      (77) 

cos  I 


COS  I 


U  =  V  -{-  w, 


in  which  w  — ;:  —  ft  in  the  case  of  direct  motion,  and  w  —  ft  —  tc 

13 


194 


THEOIIKTKAL   AHTI{()N(»MY. 


■wlicn  tilt'  distinction  of  rctrognuU'  motion  i«n(lo|»tcd;  und  wo  i-hall 
Imvo 


u"  —  u  -rz  v"  —  r, 


and,  consofinpntly, 

x«  =  r'  4-  »•'"  —  2r»-"  cos  Cu"—  u), 
K»  =  (»•"  —  r  oos  («"  —  u)Y  -V  »•'  sin' ( It"  —  «). 


W 


(78) 

(7!») 


The  vmIuc  (•('  X  derived  from  this  erjiintion  should  ngrce  witli  tliat 
already  found  from  (00). 
We  have,  further. 


or 


r  =  q  sec'  i(it  —  w), 
1  ir  ^         1 

— 7»  COS  .1  (M  —  0»)  =:        ,  _, 

1/7  Vr 


i^'  —  q  see'  ]  (?t"  —  w), 

-— ,-^-  cos  A  (n"  —  itt)  =  — -—. 


By  addition  and  subtraction,  we  get,  from  these  ecjuations, 

-     -  (cos  ]  in"  —  w)  -f-  cos  \  (n  —  w))  =  -^      +   -  -  .-, 
Vq  Vr         vr 

-'.-  (cos  A  ( u"  —  w)  —  cos  A  (i<  —  w))  =  -7^ >«, 

from  which  we  easily  derive 

2  11 

--    cos  ^  (J,  («"  -f  «)  —  w)  cos  .j  («"—  tt)  =  -7-  +  -7=-, 


Bnt 


-  --  sin  A  (.'.  (?t"  +  «)  —  m)  sin  j  (u"  —  «)  =*=  —-7- 7=. 

Vq        '   '  Vr        Vr" 

J.  _  _i  _  _i  /  ijY  _ « rr  \  ■ 

V7^V?~  t/r?'\^  r   '^^'?/' 


(80) 


and  if  wo  put 

tan  (45° +  0')-^^, 

since  \  —  will  not  differ  much  from  1,  <?'  will  be  a  smiiU  angle;  and 
we  shall  have,  since  tan  (45°  +  6')  —  cot  (45°  -{-d')  =  2  tan  26', 

>lT  +  Vf  =  2  sec  2^', 


I>AI(AIU»L1C  OltUIT. 

TluTfforo,  the  0(|Unti<>iiH  (80)  IxH^nnio 

Yq  Hill  I  in   —  It)  I'  »T 

-^  C08  J  (i(l(    +  ")  -  «") „ .7     V 

Vq  ens  |  (it    —  h)  v  rr 


195 


(81) 


from  which  the  vahics  of  y  and  to  may  1«!  fuuiul.     Then  we  nhall 
have,  for  the  lougitiKh;  of  tho  jHTihcliou 

wlicii  the  motion  is  dirctt,  and 

jr  =  ft  —  w, 

when  /  mircstrictod  exceeds  90°  and  the  distinction  of  retrograde 
motion  is  ailopted. 
It  remains  now  to  find  T,  the  time  of  perihelion  pussaf^e.   We  Imve 


=  u  —  «o, 


v"=:n"-w. 


With  the  resulting;  vahios  of  v  and  v"  we  may  find,  by  means  of 
TiiMe  VI.,  the  correspond iiij;  vahies  of  M  (which  must  Ikj  distin- 
puislicd  from  the  symbol  M  already  used  to  denote  the  ratio  of  the 
curtate  distances),  and  if  these  values  are  designated  by  J/ and  J/", 
;v(!  shall  have 

or 


T=^t~ 


M 

m 


.,      M" 

I ) 


ni 


ill  which  HI  --  -" >  and  log  C„  --  9.9601277.     When  v  is  negative,  the 

corresponding  value  of  J/ is  negative.     The  agi'cement  between  the 

two  values  of  2*  will  be  a  final  proof  of  the  accuracy  of  the  numerical 

calculation. 

The  value  of  T  when  the  true  anomaly  is  small,  is  most  readily 

and  accurately  found  by  means  of  Table  VIII.,  from   whi(!h   we 

derive  the  two  values  of  N  and  compute  the  corresponding  values 

of  T  from  the  equation 

2       - 
T=  t  —  oT  -AV'  sin  V, 

2 

in  which  log5,  =  1.6883273.     When  v  is  greater  than  90°,  we  de- 


lUU 


Til i:« »UKTI(  AL   ASTUOXOM  V 


rive  the  vali*  's  of  iV  from  tlir  tultle,  and  <'oiu|>iitc  the  coiTt'spoiulin^' 
vulucrt  of  T  from 

71.  The  ol(.'ments  q  ami  7' may  Ix'  lU'rived  dingily  from  tlio  value- 
«)f  r,  I'",  and  x,  as  jlcrivcd  from  tlu!  iMjuations  (o'J),  witliont  first 
finding  the  position  of  the  plane  of  {\\v.  orliit  an«l  tlii>  position  of  the 
orl>it  in  its  own  plane.  Tims,  the  e(|nations  (SO),  re|)lacing  u  and  )(" 
by  their  values  r   I  m  ar.d  v  |-  w",  become 


2  11 

— =  co«  \  (v"  +  vj  COS  ]  (v"  — 1>)  =  — =  -f  -7-r-. 

Addinjj;  toj^ether  the  squares  of  those,  ari.l  roducinp,  wo  get 
1       \'r^,--^,^oB},iy"-v) 


or 


9 
7  = 


sin''  ^  (v"  —  ry 
jT"sin».J(i»"  — r) 


r"  +  r  —  2  V/»V'  cos  ;1  {v"  —  v) 

Combining  this  equation  with  (59),  the  result  is 

__  rr"  sin'  j  jv"  —  v) 
9—  ,._!_/'_  X  cot  r" 

and  henec,  since  x  =  (r  +  r")  sin  ^', 

rr" 
q  =  —  sin'  \  {v"  —  v)  cot  {/. 

We  have,  further,  from  (78), 

x»  =  (r"  —  )•)'  +  4rr"  sin'  A  (v"  —  v), 

from  whii'h,  putting 

r"  —  7* 
sm  V  = , 

vc  derive 

cos  V  = sni  A  (v  —  v). 

X 


(82) 


(83^ 


(84) 
(85) 


Therefore,  the  equation  (83)  becomes 


PARABOIJC  OUIUT. 
q  =  ^(t  -f  r")  con' }/  pos'w, 


197 


l)V  moans  oC  which  tj  is  (Icrivcd  din'ctly  <V<»ni  /•,  r",  and  x,  the  value 
(if  V  Ininj;  found  l>y  means  of  the  formula  (84),  so  that  comi/  is 
|iit>iti\M'. 

When  y'  cannot  l><?  found  with  sutHcicnt  accuracy  from  the  cqua" 
tion 

wo  may  use  another  furuj.     Thus,  we  have 

r  4-  r"  4-  X 


1  +  sin  /  = 


Ji      ' 


1  —  siny': 


r 


r-{-r" 

which  give,  by  division, 

tan(45°  +  ^/)  =  JL±5±^- 
In  a  .  .i.iilur  mnuncr,  we  derive 

^  X  —  (r   —  r) 


r-f-r" 


(87) 


(88) 


Tm  order  to  fuid  th(»  time  of  perihelion  passjijre,  it  is  necessary  first 

to  derive  tlie  values  of  c  and  v".     The  ccjuations  (59)  and  (Ho)  j!;ive, 

by  luultiplication, 

tan  ^  (v"  —  I')  =  tan  /  cos  v, 


(89) 


from  which  v"  —  p  may  be  computed.     From  (82)  we  get 

\T-1 


tan  ]  iv"  +  r)  tan  j  (i;"  —  v)  = 


If  we  put 


V?+i' 


'  r 

tliis  e(|uation  reduces  to 

tan  ^  (v"  -f  v)  =  tan  (/  —  4")°)  cot .}  (v"  —  v), 
and  the  equations  (81)  give,  also, 

tan  ]  (v"  +  v)  =  cot  |  (v"  —  v)  sin  2*', 
either  of  which  may  be  used  to  find  v"  +  v. 


(90) 
(91) 


198 


THKORETICAL   ASTRONOMY, 


From  the  o(iualioiis 
cosAw 

Vq 

by  innltiplyinji;  the  first  by  sinjr"  and  the  secontl  by 
ing  tbi'  i>ro(Ui('t.s  and  rodiiciug,  wo  ca.sily  Hiul 


Vr 


co«  \v 

~Vq 


1 

V7' 


sin  Jr,  adtl- 


Hl'iico  we  have 


gin  i  ((.'"  —  r)  sin  \  v cos  \  iv"  —  v)         1 

Vq  Vr  V'/ 


1     .    ,         cotUi'"  — lO 

Vq        '  Vr 

1 


V^,."sin^(v"  — v)' 


(92) 


Vq 


cos  iv  =  — 7=, 

Vr 


which  may  be  used  to  compute  q,  v,  and  v"  when  v"  —  r  is  known. 

When  J  (r"  —  (•)  and  i(r"  •  r),  and  lience  v"  and  i-,  have  Ikh'u 
determined,  the  time  of  perihelion  passage  must  be  found,  as  ah'cady 
explained,  by  means  of  Tabic  VJ.  or  Table  VIII. 

It  is  evident,  therefore,  that  in  the  determination  of  an  orbit,  as 
soon  as  the  numerical  values  of  r,  r",  and  x  have  been  derived  irom 
the  e(iuations  (o2),  instead  of  "ompleting  the  caletdatiim  of  the  ele- 
ments of  the  orbit,  we  may  h.^vl  q  and  7',  and  then,  by  means  of 
these,  the  values  of  >•'  and  v'  may  be  computed  directly.  When  ti>is 
has  been  ett'ected,  the  values  of  n  and  n"  may  be  found  from  (3),  or 

that  of  -T,  fnmi  (34).     I'hen  we  compute  (>  by  means  of  the  first  of 

ecjnations  (70),  and  the  corrected  value  of  M  from  (33),  or,  in  the 
special  cases  already  examined,  from  the  eciuations  (37)  and  (39).  In 
this  way,  by  successive  approximations,  the  determination  of  pi.ra- 
bolic  elements  from  given  data  may  be  carried  to  the  limit  of  aecin-acy 
which  is  consistent  with  the  assumption  of  jiarabolic  motion.  In  the 
ease,  however,  of  the  ecpiations  (37)  and  (39),  the  neglected  terms 
mpy  be  of  the  second  order,  and,  consequently,  for  the  final  results 
it  will  be  necessary,  in  order  to  attain  the  greatest  possible  nccuraiy, 
to  derive 

p 


M- 


from  (15)  and  (16).  When  the  final  value  of  3/ has  been  found,  the 
determination  of  the  elements  is  completed  by  means  of  the  formuhe 
already  given. 


PAHABOLIC    ORBIT. 


199 


72.  KxAMri.R. — To  illnstriilc  tlio  application  of  the  formula'  for 
tlio  calciilatioM  of  the  paracolic  eleinciits  of  the  orl)it  of  a  ('(tinot  l)y 
a  niiinoiical  oxami)lo,  let  us  take  the  following  ob.«<C'rvations  of  the 
Fifth  Comet  of  ISO."],  made  at  Ann  Arbor: — 


Ami  Arlior  M.  T. 
18G4  Jan.  10  (J*  r,7">  20'..-) 
18  G    11    54.7 
1(5  G   35    11  .G 


19*  14™  4M)2 
19  25  2.84 
19   41    4.54 


+  34°    G'  27".4, 

36    3G  52  .8, 

+  39    41   20  .9. 


Tiii'se  places  are  referre(r  to  the  a])paront  equinox  of  the  date  and 
arc  already  corrected  for  parallax  and  aberration  by  njoaris  of 
approximate  values  of  the  jfcocentrie  distances  of  the  ct)niet.  Hut 
if  approximate  values  of  these  distances  are  not  already  known,  the 
corrections  for  parallax  and  aberration  may  be  neglected  in  the  first 
(letcrniination  of  the  apjjroximate  elements  of  the  unknown  ori)it  of 
a  comet.  If  we  convert  the  observed  right  ascensions  and  declinu- 
tioiis  into  the  corresponding  longitudes  and  latitudes  by  means  of 
O(|uations  (1),  and  reduce  the  times  of  observation  to  the  meridian 
of  \\'ashington,  wo  get 


WnshiiiRtoii  M.  T. 
18(i4  Jan.  10  7*  24"    3* 
13  G   38    37 
16  7      1    54 


297°  53'  7".6 
302  57  51  .3 
310    31  52  .3 


-f  55°  4()'  58".4, 

57    39  .35  .9, 

+  59    38  18  .7. 


Next,  we  n.'duco  these  places  by  applying  the  corrections  for  pre- 
cession and  nutjition  to  the  mean  ecpiinox  of  18G4.0,  and  reduce  the 
times  of  observation  to  decimals  of  a  dav,  and  we  have 


t  =10.30837, 
r  =  13.27(582, 
r.^.  16.29299, 


X  =.  297°  52'  51  ".1, 
r  r_:  302  57  34  .4, 
X"  =^  310    31  35  .0, 


,?  =.  +  55°  46'  58".4, 
ir  =  57  39  35  .9, 
fi"=.  +  5d    38  18  .7. 


For  the  same  times  we  find,  from  the  American  Xaiitical  Almanac, 


O  -=290°  6'27".4, 
O'  =.  293  7  .57  .1, 
0"  =  29G    12  15  .7, 


log  R  =  9.9927G3, 
logie'  =9.992s:i(), 
log /r  =  9.99291(5, 


which  are  referred  to  the  mean  equinox  of  1864.0.     It  will  gene- 
rally be  sufficient,  in  a  first  ai)proxiniation,  to  use  logarithms  of  five 
decimals;  but,  in  order  to  exhibit  the  caleuhition  in  a  more  complete 
form,  \i  _'  shall  retuin  six  places  of  decimals. 
Since  the  intervals  are  very  nearly  o(iual,  we  may  assume 


!; 


\''"' 


200 

Then  we  have 
M 


TIIEORETICAI.   ASTRONOMY. 


n 


?l 


T 


'N"' 


and 


/"  —  <;     tanks'  sin  (X  —  Q')^—  tan/5  sin  (^-'^-^G^ 
t'  —  t  '  tan, J"  sin  (A'  —  ©')  —  tan,*'  sin  (A"  —  0')' 

ff  mi  (G—Q)=:  R"  .sin  (©"  —  O), 

(J  cos  (  Ct'  —  0)  --'--  R"  co.s(  0"  —Q)  —  R; 

h  cos  :  coaiH—k")  =  M—  cos  (a"—  A), 

/i  cos  :  sin  (H—).")  ^  sin  (a"  —  ;.), 

A  sin  C  =  J/  tan  ,5"  —  tan ,? ; 

from  which  to  find  M,  G,  f/,  II,  ^,  and  h.     Thus  we  obtain 


log  iV=.  9.821)827, 

G  ==  22°  58'  1".7, 
log5r=^!).019<J18, 


94°  24'    1".8, 
:  =  —  40    28  21  .9, 
log  A  ==9.688532. 


J" 


cos  /5 


0.752,  it  appears  that  the  comet,  at  the  time 


Since  --  =  M 

J  cos  ,1" 

of    the.se    observations,    was    rapidly    approaching    the   earth.     The 

(puulrants  in  which  G  —  0  and  11—)."  must  be  taken,  are  deter- 

niiiKHl  by  the  condition  that  r/ and  /leos^  must  always  be  positive. 

TIjc  value  of  .1/"  should  be    becked  In'  duplicate  calculation,  since  an 

error  in  this  will  not  be  exhibited  until  the  values  of  /'  and  /?'  are 

computed  from  the  resulting  elements. 

Next,  from 

cos  ^  —  cos  ,3  cos  U~  Q ),  cos  +''  =  cos  /S"  cos  (/."  —  0"), 

cos  ^p  =  cos  C  cos  (  G  —  // ), 

we  compute  cos  -i^,  co.s  ^",  and  cos  f ;  and  then  from 

g  sin  <p      =  A,  h  cos  /?  =  b, 

Rs'm-^    ==  B, 

R"smV'  =  B", 
g  cos  tp  —  hR  cos  +  =  c,  g  cos  <f>  —  b"R"  cos  i"  =  c", 

we  obtain  A,  B,  B",  &c.  It  will  generally  be  sufficiently  exact  to 
find  sin  ^  and  sin  ^l/"  from  cos  ^  and  cos\//";  but  if  more  accurate 
values  of  '^  and  i//"  are  required,  they  may  be  obtained  by  means  of 
the  ecpiations  ^42)  and  (43).     Thus  we  derive 

log  A  =^  9.006485,        log  B  =  9.91 2052,        log  B'  =  9.933366, 
log  6  =  9.438524,  log  6"  =  9.562387, 

c  ==  —  9.125067,  c"  =  —  0.150562. 


h  cos  ,5"       ,  „ 


NUMEP.ICAL  EXAMPLE. 


201 


Then  wc  have 

T'  =  k(t"-t), 

2t' 


2/ 


X  = 


Vr-\- 


jil^y 


V(l+')" 


+  5^ 


from  which  to  find,  by  successive  trials,  the  vahics  of  r,  r",  and  x, 
that  of  //  being  found  from  Table  XI.  with  the  argument  r^.  First, 
we  assume 

log  X  =  log  t'v2  =  9.163132, 

and  with  this  we  obtain 

log  r  =  9.913895,        log  r"  r=  9.938040,        log  (r  +  r")  =  0.227165. 

This  value  of  log(r  +  r")  gives  rj  =  0.094,  and  from  Table  XI.  we 
find  log/i  -  0.000160.     Hence  we  derive 

log  X  =  9.200220,         log  r  =  9.912097,        log  r"=  9.935187, 
log(7-  +  r")  =  0.224825. 

Repeating  the  operation,  using  the  last  value  of  log(r  +  r"),  we  get 

log  X  =:  9.201396,        log  r  ==  9.912083,        log  r"  =  9.935117, 
log(r  +  r")  =  0.2;^783. 

The  correct  value  of  log(j*  +  r")  may  now  be  found  by  moans  of  the 
equation  (67).  Thus,  we  have,  in  units  of  the  sixth  decimal  place  of 
the  logarithms, 

a  =  224825  —  227165  =  —  2340,        a'  =  224783  —  224825  =  —  42, 

and  the  correction  to  tlie  last  value  of  log{r  -\-  r")  becomes 


a' 
a'  —  a 


0.8. 


Therefore, 

log(j-  +  r")  =  0.224782, 

and,  recomputing  rj,  n,  x,  r,  and  r",  we  get,  finally, 

log  X  =  9.201419,        logr  =  9.912083,        log  r"  =  9.935116, 
log(r  +  r")-=  0.224782. 

The  agreement  of  the  last  value  of  log(r  +  '"")  with  the  preceding 
one  shows  that  the  results  arc  correct.     Further,  it  appears  from  the 


202 


THEORETICAL   ASTRONOMY. 


values  of  )•  !ui<l  r"  that  tlio  comc't  had  pat^scd  its  |>orihc'lion  and  was 
rt'ccdin^  IVoin  the  sun. 

iJy  nn-ans  of  tho  vahies  of  r  and  /■"  wi-  nnj^ht  oonipnto  ai)|)r(».\i- 

niati'  vahics  of  /■'  and    .    from  the  equations  (.'JO)  and  (•»!),  and  then 

((I  ^  f. 

a  more  approximate  vahie  of     „  from  (28),  tliat  of  ^,^  being  found 

from  (;{2).     IJiit,  sinee  r'  differs  but  little  from   Ji',  the  ditt'erenoo 

between     „  and  ,rr,;  is  vcrv  small,  so  that  it  is  not  neeessarv  to  eon- 
H  J\ 

sider  the  second  term  of  the  seeond  member  of  the  etpiation  {'•i'-)); 

and,  since  the  intervals  are  very  nearly  e(iual,  the  error  of  the  as- 

sunjption 

n  T 

is  of  the  third  order.  It  should  be  observed,  however,  that  an  error 
in  the  value  of  J/ affects  //,  ^,  li,  and  luMice  also  A,  h,  h",  c,  and  r;", 
and  the  resnilint;  value  of  fi  may  be  affected  l)y  an  error  which  con- 
siderably exceeds  that  of  ^^.  It  is  advantap'ous,  therefore,  to  select 
observations  which  furnish  intervals  as  nearly  ecjnal  as  possil)le  in 
order  that  the  error  of  J/  may  be  small,  otherwise  it  may  become 
necessary  to  correct  J/^and  to  repeat  the  calculation  of  f,  r",  and  x. 
We  may  also  compute  the  perihelion  distance  and  the  time  of  I'cri- 
helion  ])assafjje  from  /•,  /•",  and  J£  by  means  of  the  e(piations  (8(J),  (89), 
and  (01)  in  connection  with  Tables  VI.  and  VIII.  Then  r'  and  c' 
may  be  computed  directly,  and  the  eomp'"te  expression  for  M  may 
be  employed. 

In  the  first  determination  of  the  elements,  and  espe<  lally  when  the 
corrections  for  parallax  and  aberration  have  been  neglected,  it  is  un- 
necessary to  attem[)t  to  arrive  at  the  limit  of  accuracy  attainable, 
since,  when  approximate  elements  have  been  found,  the  observations 
mav  hv  more  convenientlv  reduced,  and  those  which  ini-lude  a  lonyior 
interval  may  be  used  in  a  more  complete  calculation.  Hence,  as  soon 
an  r,  r",  and  x  have  been  found,  the  curtate  tlistances  are  next  deter- 
mined, and  then  the  elements  of  tlie  orbit.     To  find  ft  and  //',  we 

have 

rf  =  + 0.122395, 

the  positive  sign  being  used  since  x  is  greater  than  g,  and  the  forinulte 


h         ' 


give 


log />  1=9.480952, 


log//' =-9.310779. 


NIMKRICAr-    KXAMPLE. 


203 


From  tlicso  values  of  (>  and  //',  it  appt-ars  ihat  the  comet  was  very 
luar  the  earth  at  tite  time  of  the  observations. 

The  heliooeiitrie  j)hurs  are  then  fonnil  l)y  means  of  the  equations 
(71)  ami  (72).     Thus  we  obtain 

/  r..:  106°  40'  oO".r),        6  =--  +  ;}:}°    r  10".0,        logr  =  9.0120^2, 
r^lVl    ;U     I)  .}>,        Z»"-^  +  28    05     5.8,         Iogr"=U.!»;5511(i. 

Till'  agreement  of  these  values  of  /•  and  r"  with  those  previously 
liiund,  cheeks  the  accuracy  of  the  calculation.  Fiu'ther,  since  the 
litliocentric  l(»nu;itu(lcs  are  increasing,  the  motion  is  dirvof. 

The  longitude  of  the  ascending  node  and  the  inclination  of  the 
orbit  may  now  be  found  by  means  of  the  ecpiations  (74),  (75),  or  (7U); 
and  we  get 

Si  ==  304'^  43'  11".5,  i  =  04°  31'  21".7. 

The  values  of  w  and  n"  are  given  by  the  formula; 


tan  u 


tan(/— Si) 


cos  I 


tan  It" 


tan(/"—  Q,) 


COS! 


('  and  /  —  Si  being  in  the  same  quadrant  in  the  case  of  direct  motion. 
Thus  we  obtain 

u  =  142"^  52'  12".4,  u"  =  153°  18'  49".4. 

Then  the  equation 

x»  =  (}•"  —  r  cos ( h"  -  n)y  +  r'  sin'  («"  —  u) 
log  x  =  9.201423, 


gives 


and  the  agreement  of  this  value  of  x  with  that  previously  found, 
proves  the  calculation  of  Q,,  i,  u,  and  u". 
I'Vom  the  e({uations 


*    r- 


wo  get 


tan  (45°  -}-<?')-  \/~-, 

1      •    ,i,  /■  f,-  ,     s         -v  tan 20' 

V  q  sm  }  (u  —  M )  |/  rr 

1          1  / 1  /  "  1     \         \                 sec  2<i'' 
— .^  cos  J  (.J  (tt '  4-  u)  —  m)  — -;p— , 

V  q        '  cos  \  iu" —  u)  vrr" 


ef  ^  0°  22'  47".4,        a,  =  115°  40'  6".3,        log  q  =  9.887378. 

Hence  we  have 

TT  =  w  +  Si  =  60°  23'  17".8, 


r"  =  u"— «>  =  37<'38'43".l. 


204  THEORETICAL  ASTRONOMY. 

and 

v  =  u  —  m  =  2V  12'6".l, 
Then  we  obtain 

log  m  =  9.Jt()01277  —  3  log  ry  -^  0.129061, 
and,  corresponding  to  the  values  of  v  and  r",  Table  VI.  gives 

log  M=  1.267163,  log  M"  =  1.4241o2. 

Therefore,  for  the  time  of  perihelion  pa.ssage,  wo  have 


T^t   ~  '^'^^  =-<  — 13.74364, 
m 


and 


T^t"- 


M" 


m 


:/"— 19.72836. 


The  first  value  gives  T^  1863  Dec.  27.56473,  and  the  second  gives 
T=  Dec.  27.56463.  The  agreement  between  these  results  is  the  final 
proof  of  the  calculation  of  the  elements  from  the  adopted  value  of 

p 
If  we  find  T  by  means  of  Table  VIII.,  we  have 

log  iV^^  0.021616,  logiV"=r.  0.018210, 

and  the  equation 

9  2 

T=t  —  -^Nri  sin  v  =  t"  —  -^  N"r"^  sin  v", 

in  which  h>g.^,=-- 1.5883273,  gives  for  T  the  values  Dec.  27.56473 

and  Dec.  27.56469. 

Collecting  together  the  several  results  obtained,  we  have  the  fol- 
lowing elements : 

T  =  1863  Dec.  27.56471  Washington  mean  time. 

ff   =   60°23'17".8)    ^  ,.    . 

i-»       on«     ^o  11    -  I    t-cliptic  and  Mean 

log  7  =  9.887378. 

Motion  Direct 

73.  The  elements  thus  derived  will,  in  all  cases,  exactly  represent 
the  extreme  places  of  the  comet,  since  these  only  have  been  used  in 
finding  the  elements  after  f)  and  f}"  have  been  found.     If,  by  means 


NUMERFCAI-    r^XAMPLES. 


205 


of  tli('«o  elomonts,  wc  compute'  n  and  »",  and  correct  the  value  of  ^f, 
the  elements  which  will  then  he  ohtained  will  a|)j)roxiniate  nearer 
the  tnu!  values;  and  each  successive  correction  will  ftu'nish  more 
ai'carate  results.  When  the  adopted  value  ol'  .1/ is  exact,  the  result- 
iiijf  elements  must  by  calculation  rej)roduce  this  value,  and  since  the 
(diiiputed  values  of  ?.,  )'\  ,9,  and  (i"  will  he  the  same  as  the  observed 
values,  the  «'omputed  values  of  /'  and  [i'  nujst  bo  such  that  when 
suhstituted  in  the  e(piation  for  J/,  the  same  result  will  be  obtained 
a-  when  the  observed  values  of  //  and  (i'  an;  used.  liut,  according 
til  the  e(piations  (l.'J)  and  (14),  the  value  of  .l/(U'pends  only  on  the 
inclination  to  the  ecliptic  of  a  great  circle  |)assinir  through  the  places 
of  the  sun  and  comet  for  the  time  ^',  and  is  independent  of  the  angle 
at  the  earth  between  the  sun  and  comet.  Hence,  the  spherical  co- 
onliiiates  of  any  point  of  the  groat  circle  joining  these  places  of  the 
sun  and  conu't  will,  in  connection  with  those  of  the  extrcMue  j)laces, 
give  the  same  value  of  J/,  and  when  the  exact  value  of  M  has  been 
iwd  in  deriving  the  elements,  the  computed  values  of  //  and  ^•i'  must 
jrive  the  same  value  for  \c'  as  that  which  is  obtained  from  observa- 
tion. But  if  we  represent  by  ^'  the  angle  at  the  earth  between  the 
sun  and  comet  at  the  time  i\  the  values  of  ^'  derived  by  observation 
and  by  computation  from  the  elements  will  differ,  uidess  the  middle 
jjiace  is  exactly  represented.  In  general,  this  ditferenee  will  be  small, 
aud  since  ic'  is  constant,  the  equations 


cos  ij  =  cos  (5'  cos  (A' 
sin  4'  cosn»'  ■=  cos  S  sin  (A'  - 
sin  V  sin  w'  =  sin  {i\ 


©'), 


(93) 


give,  by  differentiation, 


(94) 


cos  ^  d)!  =  cos  n''  sec  ,S'  d^' , 

d-i'  =  sin  v/  cos  (A'  —  ©')  d\\ 
From  these  we  got 

cos  [-i'  dk'  _  tan  U'  —  OO 
d^      ~        sin/       ' 

which  expresses  the  ratio  of  the  residual  errors  in  longitude  and 
liititiide,  for  the  middle  place,  when  the  correct  value  of  M  has  been 
u>e(l. 

Whenever  these  conditions  are  satisfied,  the  elements  will  be 
correct  on  the  liypothcsis  of  parabolic  motion,  and  the  magnitude 
of  the  final  residual::,  in  the  middle  place  will  depend  on  the  deviation 
of  the  actual  orbit  of  the  comet  from  the  parabolic  form.     Further, 


20G 


TII  RORKTIf  A  L   ASTRONOMY. 


wlit-'ii  c'lf'inontH  li!iv(-  Ikm'ii  (Icrivcd  from  a  value  of  3/  which  has  iidt 

been  finally  convctod,  if  we  compute  /'  and  (i'  by  means  of  those 

elements,  and  then 

fnn  ,5' 


tan  ,^ 

tan  IV  =   .—     -~_-y^,, 
8m(A'  —  O'j 


tlie  comparison  of  this  value  of  tan«''  with  that  given  by  obs(M'va- 
tion  will  show  whether  anv  further  correction  of  M  \n  neeessarv,  and 
if  the  difference  is  not  greater  than  what  may  be  due  to  iniavoidalilc 
errors  of  ctileulation,  we  may  regard  M  its  (jxact. 

To  compare  the  elements  obtained  in  the  case  of  the  example 
given  with  the  middle  place,  we  find 


w'r^32°3r  13"..5, 
Then  from  the  erpiations 


It 


148°  11'  19".8, 


log/-'  =  9.9228:]6. 


we  derive 


tan  (r  —  JJ  )  =  cos  i  tan  u', 

tan  b'  =  tan  i  sin  il'  —  J^), 


r  =  109°  46'  48".3,  b'  =  28°  24'  56".0. 

By  means  of  these  and  the  values  of  0'  and  R',  we  obtain 

>l'  =  302°  57'  41".l,  /5'  =  57°  39'  37".0 ; 

and,  comparing  these  results  with  the  observed  values  of  X'  and  ,i', 
the  residuals  for  tlie  middle  place  are  found  to  be 

Com  p.  —  Obs. 
cos  ;/  aA'  =  +  3".(},  A/3  =  +  l".l. 

The  ratio  of  these  remaining  errors,  after  making  due  allowanro  hx 
unavoichible  errors  of  calculation,  shows  that  the  adopted  value  of 
M  is  not  exact,  since  the  error  of  the  longitude  should  be  less  tlnin 
that  of  the  latitude. 

The  value  of  w'  given  by  observation  is 

log  tan  ?t)'  =  0.966314, 

and  that  given  by  the  computed  values  of  )J  and  /?'  is 

log  tan  xo'  =  0.966247. 

The  difference  being  greater  than  what  can  be  attributed  to  errors  of 
calculation,  it  appears  that  the  value  of  M  requires  further  cor- 


Xr.MKRICAL   EXAMPIii^. 


207 


rection.     Since  the  diflbrcnee  is  siduII,  wo  may  dorivc  tlio  oorrcrt 

viiliit'  of  M  l)y  u.siiig  tlio  same  ussutued  value  of  -,,.  and,  instead  of 

tlie  value  of  tan  w'  derived  from  ol)Hervation,  a  value  difrerinj^  as 

imicli  from  this  in  a  contrary  direction  as  the  computed  value  differs. 

Tims,  in  the  )>resent  example,  the  computed  value  of  lo^  tan  w'  is 

0.<H)()(K57  less  than  the  observed  value,  and,  in  finding  the  new  value 

of  M,  we  must  use 

log  tanit;' =  0.966381 

in  oonijmting  /?„  and  fij'  involvc<l  in  the  first  of  equations  (14).  If 
the  first  of  e([Uations  (10)  is  employed,  we  must  use,  instead  of  tan^?' 
Q.)i  derived  from  observation, 


or 


tan  [^  =  tan  w'  sin  (A'  —  ©'), 

log  tan  /5'  ^  0.966381  +  log  sin  (A'  —  0')  =  0.198559, 

the  observed  value  of  //  being  retained.     Thus  we  derive 

log  i»f=  9.829586, 

and  if  the  elements  of  the  orbit  are  computed  by  means  of  this 
value,  they  will  represent  the  middle  place  in  accordance  with  the 
condition  that  the  diflerenee  between  the  computed  and  the  observed 
value  of  tan  w'  shall  be  zero. 

X    system  of   elements    computed    with    the    same    data    from 
log  M  =  9.822906  gives  for  the  error  of  the  middle  place. 


C.-O. 

cos/5'a;/  =  — 1'26".2, 


A/9' 


40".l. 


If  we  interpohitc  by  means  of  the  residuals  thus  found  for  two  values 
of  M,  it  appears  that  a  system  of  elements  computed  from 

log  iJf=  9.829586 

will  almost  exactly  represent  the  middle  place,  so  that  the  data  are 
completely  .satisfied  bv  the  hypothesis  of  parabolic  motion. 
The  equations  (34)  and  (32)  give 

log  —  =  0.006955,  log  -^  =  0.006831, 


and  from  (10)  we  get 

log  ilf' =  9.822906, 


Iogif"  =  9.663729„. 


208 


TII KOUKTICA I,    AHTIK »N()M V. 


'i'lioii  l»y   niciui.s  of  the  i'(|Uution  (33)  we  derive,  for  the  eonvftoil 

vuhie  of  M, 

loj,'.V^9.82J)r)82, 

wliieh  (lilli  rs  only  in  the  .sixth  dwiiniil  phu.'t?  from  the  result  ohtaincd 
by  vtiryiii;j  tainr'  and  retaining  the  approxitiuite  values  —  =-t»=^\v7,- 

74.  When  the  approxiiiiiite  elements  of  the  orbit  of  a  eomet  iiit' 
known,  they  may  be  eorreeted  by  usinjf  observations  which  iiicludr 
a  hinder  interval  of  time.  The  most  convenient  method  of  en'ectiiig 
this  correction  is  by  the;  variation  of  the  j^eoeentrie  distanee  for  tlio 
time  of  one  of  the  extreme?  observations,  and  the  fornudiu  which 
may  be  derived  f!tr  this  purpose  are  applicaltle,  without  modilicatioii, 
to  any  case  In  wliich  it  is  possible  to  determine  the  elements  of  the 
orbit  of  a  comet  on  the  sup})osition  of  motion  in  a  parabola.  Since 
there  are  (»nly  five  elements  to  be  determincil  in  tlu;  ease  of  parabolic 
motion,  if  the  distanee  of  the  eomet  from  the  earth  correspond inj;  to 
the  time  of  one  complete  observation  is  known,  one  additional  com- 
plete observation  will  enable  us  to  tind  the  elements  of  the  orbit. 
Theretbre,  if  the  elements  are  computed  which  result  from  two  or 
more  assumed  values  of  J  dill'eriuf;  l)Ut  little  from  the  correct  value, 
i)y  eompari.son  of  intermediate  obsei  ,ations  with  tlie.se  ditterent  sys- 
tem.s  of  elements,  we  may  derive  that  value  of  the  jroocentric  distance 
of  the  eomet  for  which  the  resulting  elements  will  best  rcpre.sent  the 
observations. 

In  order  that  the  formula;  may  be  applicable  to  the  case  of  any 
fundamental  plane,  let  us  consider  the  equator  a.s  this  plane,  and, 
supposing  the  data  to  be  three  complete  observations,  let  A,  A',  A" 
be  the  right  a.seension.s,  and  1),  I)',  I)"  the  declinations  of  the  sun 
for  the  times  /,  t',  I".  The  co-ordinates  of  the  first  place  of  the  earth 
referred  to  the  third  are 

X  =  R"  cos  ly  cos  A"  —RcosD  cos  A, 
y  =  JH'  cos  IJ"  sin  A"  —  li  cos  Ds'niA, 
z  =  R"  sin  ly  —R  sin  D. 

If  we  represent  by  «/  the  chord  of  the  earth's  orbit  between  the  places 
for  the  first  and  third  observation.s,  and  by  G  and  K,  respectively, 
the  right  ascension  and  declination  of  the  first  place  of  the  earth  as 
seen  from  the  third,  we  .shall  have 

x  =  g  cosK  cos  G, 
y=zg  cosK  sin  G, 
z  =  g  sinK, 


VAIUATION   OF  TUK   OKOCENTRIC   DISTANCE. 
and,  {•oiispcuK'iitly, 


209 


fj  COH  A'  cos  {a—A)=-  Ji"  0(W  />"  cos  (A"  A)  —  Ii  COB  /), 

(J  CM  A'  sin  (  G  —  A)  ■----.--  It"  cos  />"  sin  ( .1"      A ),  0)(y ) 

(/  sin  A'  ^  Ti"  sin  //'      Ji  sin  7^ 

t'roMi  which  //,  A',  and  (f  may  l)c  found. 

It'  we  dcsii^nati!  hy  .r„  i/„  z,  the  co-ordiimtcs  of  tlu>  first  place  of 
tlic  conu't  referred  to  the  third  place  of  the  earth,  we  shall  have 

x,=  J  cos ')  cos  a  -{-  y  cos  A'  cos  G, 
y,  =  J  cos  "  HJn  o  -{-  (/  cos  A'  sin  G, 
z,  =  J  sin  o  -\-  y  sin  A'. 


Lot  us  now  put 


und  we  get 


a-,  ^  h'  cos  :'  cos  //', 
y,  =  h'  cos  C'  sin  W, 
2,  =  h'  sin  C', 


/i'  cos  C'  cos(H'  —  O)  =  J  cos 'T  cos  (»  —  ^)  +  i7  cos  K, 
h'  cos  :'  sin  (//'  -  G)  =  J  cos  -1  sin  (a  —  G), 
h'  sin  C  =  J  sin  >>  -{-  g  sin  A", 


(97) 


from  which  to  determine  H',  !^',  and  h'. 

If  we  represent  by  <p'  the  angle  at  the  third  place  of  the  earth 
Ix'twcen  the  actual  first  and  third  places  of  the  comet  in  space,  we 
ol)tiiin 

cos  V''  —  cos  C'  cos  //'  cos  <5"  cos  a"  -\-  COS  C'  slu  //'  COS  o"  sin  a"  -|-  sin  C'  sin  'J", 


or 


COS  <p'  =  COS  C'  cos  <J"  COS  (tt"  —  //')  -|-  sin  C'  sin  (J" ; 
and  if  we  put 


K.V6) 


tills  becomes 

Tlicn  we  shall  have 


e  sin/=  sin  '5", 

e  cos/=  cos  '5"  cos  (a"  —  H') 

cos  <p'  =  e  cos  (?'  — /). 


x»  =  A'»+ J"»— 2/i'J"cos/ 


or 


x»  =  (  J"  —  A'  cos  <py  +  /i''  sin'  /, 


(99) 


(100) 


in  which  J"  is  the  distance  of  the  comet  from  the  oarth  corrcspond- 
injf  to  the  last  observation.     We  have,  also,  from  equations  (44)  and 


r*  =  ( J  —R  cos  4)»     +  R*  sin*  ^, 
(J"  —  i?"  cos  ^y  +  ii"'  sin'  +", 


h"» 


(101) 


14 


210 


TIIKOUKTICAL   ASTflOXOMY. 


ill  which  -v//  is  the  aiij^lo  at  fho  t-arth  hctwrcn  tho  sun  and  oomot  at 
the  tiimt  /,  ami  ^"  the  same  angle  ut  the  time  i".  To  liiul  thtir 
vahies,  we  hav(! 


coH  ^  -—  C08  D  cos  S  ros  (o  —  yl)  -f-  **'n  ^^  s'"  '^ 
cos  4"^-.  cos  ])"  cos  ')"  co8(tt"—  ^")  +  Hill  I)"  sin  'J", 


rio'2) 


Avhicli  may  he  still  further  reduced  hy  the  intriKluction  of  auxiliary 
angles  as  in  the  ease  of  ec^uation  (J'8). 
Let  us  now  put 


/i'siny'=a 

It  sin  ^  =  B, 

H"  sin  i"  -=  Ji", 


h'  cog  y>'  =  c, 

R  cos  ■^  --  i, 

ii"  cos  +"  =  6", 


(103) 


and  we  shall  have 


(104) 


These  equations,  together  with  (56),  will  enable  us  to  determine  J" 
by  successive  trials  when  J  is  given. 

We  may,  therefore,  assume  an  approximate  value  of  J"  by  means 
of  the  approximate  elements  known,  and  find  r"  from  the  last  of 
these  equations,  the  value  of  r  having  been  already  found  from  the 
assumed  value  of  J.     Then  x  is  obtained  from  the  equation 


2r'__ 

Vr  +  r" 


Ih 


fi  being  found  by  moans  of  Table  XI.,  and  a  second  approximation 
to  the  value  of  J"  from 


J"  =  c±v/x'^— C" 


(105) 


The  approximate  elements  will  give  J"  near  enough  to  show  whether 
the  upi)er  or  lower  sign  must  be  used.  With  the  value  of  J"  thus 
found  we  recompute  r"  and  X  as  before,  and  in  a  similar  manner  find 
a  still  closer  approximation  to  the  correct  value  of  J".  A  few  trials 
will  generally  give  the  correct  result. 

When  J"  has  thus  been  determined,  the  heliocentric  places  are 
found  by  means  of  the  formulee 


r  cos  b  cos  (I  —  A)  =  J  cos  S  cos  (a  —  A)  —  R  cos  D, 
r  cos  b  sin  {I  —  A)  =  J  cos  S  sin  (»  —  A), 
r  sin  6  =  J  sin  5  —  R  sin  Z>; 


(106) 


VAIUATION   OF  TIIK   (iECM'KNTUIC   DIHTAXCE. 


211 


/'coHrcoHfr-    A") 

/•"n.s/>".Hin(r-vl") 
/•"  sin  //' 


J"  cos  A"  cos  ( a"  -  A")  —  n"  cos  /)", 

J"  COH  <)"  m\  ( a"       A"),  (107) 

J"  sin  -5"  -  i{"  sin  />", 


in  wliicli  /),  h",  /,  /"  arc  tlic  heliocentric  .splicriciil  co-ordinates  rc- 
lirrcd  to  the  equator  as  the  t'nndaniental  phiiie.  The  vahie-*  of  r  and 
>•"  tJnind  IVoin  these  ecpmtions  must  agree  with  those  obtained  I'roni 

The  eh'inents  of  the  orbit  may  now  l)e  determined  by  means  of  the 
(■(|iiations  (7")),  (77),  and  (HI),  in  connection  with  Tabh-s  VI.  and 
VI I  J.,  as  abT:idy  explained.  The  elements  thns  derived  will  be  re- 
il'ircd  to  the  eijii:itor,  or  to  a  plane  passinjr  through  the  centre  of  the 
sun  and  parallel  to  the  earth's  equator,  an<l  they  may  be  transformed 
into  those  for  the  celiptie  us  the  fundamental  plane  by  means  of  the 
wjuations  (109),. 

7o.  With  the  resulting  elements  wc  eomptite  the  place  of  tl.e  comet 
for  the  time  I'  and  compare  it  with  the  corresponding  observed  place, 
and  if  we  denote  the  eomimted  right  ascension  and  deoliuation  by  oty' 
and  «(,',  respectively,  we  shall  have 


o  -f-  a  =^  o, 


■o» 


.r  +d'  =  >\', 


in  which  a'  and  d'  denote  the  differences  between  eom[»utation  and 
olworvation.  Next  we  assume  a  second  value  of  J,  which  we  repre- 
sent by  J  -j-  oJ,  and  compute  the  corresponding  system  of  elcmenb*. 
Then  we  have 

a"  and  d"  denoting  the  differences  between  computation  and  obser- 
vation for  the  second  system  of  elements.  We  also  compute  a  third 
system  of  elements  with  the  distance  J  —  uJ,  and  denote  the  ditfer- 
cnces  between  computation  and  observation  by  a  and  d;  then  we  shall 

have 

«=/(J-<5J),  a'=/(J),  a"=/(J  +  ,5J), 

and  similarly  for  d,  d',  and  d".  If  thtso  three  numbyrs  are  exactly 
represented  by  the  expression 


X 


m  +  n-^-^  + 


ijjf' 


in  which  J  +  .r  is  the  general  v  "'ue  of  the  argument,  since  the  values 
of  u,  a',  and  a"  will  be  such  th  t  the  third  differences  may  be  neg- 
lected, this  formula  may  be  assumed  to  express  exactly  any  value  of 
the  function  corresponding  to  a  value  of  the  argument  not  differing 


212 


THEORETICAL   ASTRONOMY. 


imicli  from  J,  or  within  the  limits  x  =  —  SJ  and  x  =  -f-  SJ,  the  as- 
sunu'd  vahios  J  —  dJ,  J,  and  J  -}-  (J J  being  ,so  taken  that  the  correct 
value  of  J  shall  be  either  within  these  limits  or  very  nearly  so. 
To  find  the  coefficients  7/i,  v,  and  o,  we  liave 


m  —  ?i  -|-  0  =  a, 


whence 


m 


n 


m  ^^  a , 


K«"-«). 


m  -\-  n  -\-  0 


o  =  A(«"+fl) 


Now,  in  order  that  the  middle  place  may  be  exactly  represented  in 
right  ascension,  we  must  have 


''(*!,)'+"(,;j)+"'=0' 


from  whicli  wc  find 


X 


-  2o  ^" 


l/?j,*  —  4mo)  =  p, 


or 


x—p,U  =  0. 


In  the  same  manner,  Jie  condition  that  the  middle  place  shall  I)e 
exactly  represented  in  declination,  gives 

In  order  that  the  orbit  shall  exactly  represent  the  middle  place,  l)otli 
conditions  must  be  satisfied  simultaneously;  but  it  will  rarely  happen 
tliat  this  can  i)e  efiected,  and  the  correct  value  of  x  must  be  found 
from  those  obtained  by  the  sej)arate  conditions.  The  arithmetical 
mean  of  the  two  values  of  x  will  not  make  the  sum  of  the  squares 
of  the  residuals  a  mininuun,  and,  therefore,  give  the  most  probable 
value,  uidess  thu  \anation  of  cos  5' Aa',  for  a  given  increment  as- 
signed to  J,  is  the  same  as  that  of  aS'.  But  if  we  denote  the  value 
of  X  for  which  the  error  in  a'  is  reduced  to  zero  by  x',  and  that  for 
Avhich  Ao'  =  0,  by  x",  the  most  probable  value  of  x  will  be 


X- 


n^x'  4-  n'Kv" 


(108) 


in  which  n  ^=^-  h{(i" —  a)  and  n'  =  l{(l" —  d).  It  should  be  observed 
that,  in  order  that  the  ditferences  in  right  ascension  and  declination 
shall  have  equal  influence  in  determining  the  value  of  x,  the  values 
of  a,  a',  and  <i"  must  be  multiplied  by  cos<J'.  The  value  of  3 J  is 
most  conveniently  expressed  in  units  of  the  last  decimal  place  of  the 
logarithms  employed. 


NUMERICAL   EXAMPLE. 


213 


If  the  elements  are  already  known  so  approximately  that  the  first 
assumed  value  of  J  (liliei*s  so  little  from  the  true  value  that  the 
sot'oud  diiforences  of  the  residuals  may  he  ne<flectcd,  two  assumptions 
in  regard  to  the  value  of  J  will  suffice.  Then  we  shall  have  o  -  0, 
and  hence 


m 


a. 


n  ==:  O     (I  , 


The  condition  that  the  middle  place  shall  be  exactly  represented, 
gives  the  two  equations 


(«"  —  a')  X  +  a'<U  =  0, 
l^d"  —  d')  X  +  d'U  =  0. 


(109) 


The  combination  of  these  equations  according  to  the  method  of  least 
S(|uarcs  will  give  the  most  probable  value  of  .r,  namely,  that  for 
which  the  sum  of  the  squares  of  the  residual.s  will  be  a  mininuun. 

Having  thus  determined  the  most  probable  value  of  x,  a  final 
system  of  elements  (computed  with  the  geocentric  distance  J  -\-  x, 
corresponding  to  the  time  /,  will  represent  the  extreme  places  exactly, 
and  will  give  the  least  residuals  in  the  middle  pla"  •  consistent  with 
the  supposition  of  parabolic  motion.  It  is  further  evideat  that  we 
may  use  any  number  of  intermediate  places  to  correct  the  assumed 
value  of  J,  each  of  which  will  furnish  two  equations  of  condition 
for  the  determination  of  x,  and  thus  the  elements  may  be  found 
which  will  represent  a  series  of  observations. 

7G.  Example. — The  formula)  thus  tierlved  for  the  correction  of 
a})proximate  parabolio  elements  by  varying  the  geocentric  distance, 
are  applicable  to  the  case  of  any  fundamental  ])lane,  provided  that 
a,  0,  A,  I),  etc.  have  the  same  signification  with  respect  to  thiii  plane 
that  they  have  in  reference  to  the  equator.  To  illustrate  their 
immcrical  application,  let  us  take  the  following  normal  i)lacc3  of 
the  (Jreat  Comet  of  1858,  wiiicli  were  derived  by  comparing  an 
ei)lR'm?ris  with  several  observations  made  during  a  few  divys  before 
and  after  the  date  of  each  normal,  and  finding  the  mean  difference 
between  computation  and  observation: 


Washington  M.  T. 
1858  June  11.0 
July  13.0 
Aug.  14.0 


141°  18'  30".9 
144  32  49  .7 
152    14  12  .0 


+  24°  46'  25".4, 

27    48     0  .8, 

+  31    21  47  .9, 


which  arc  referred  to  the  apparent   equinox  of  the  date.     These 
places  arc  free  from  aberration. 


214 


THEORETICAL    ASTRONOMY. 


We  shall  take  the  ecliptic  for  the  fiindainental  plane,  and  con- 
verting these  rifijht  ascensions  and  declinations  into  longitudes  and 
latitudes,  and  reducing  to  the  ecliptic  and  mean  equinox  of  1858.0, 
the  times  of  observation  being  exi)ressed  in  days  from  the  beginning 
of  the  year,  we  get 


t  =162.0, 
e  =  194.0, 
t"  =:  226.0, 


A  =  135°  51'  44".2, 
X'  =  137    39  41  ,2. 

/'  =--- 142    51  31  .8, 


/5  =+    9°    6'57".8, 
lY  =      12    55    9  .0, 

fi"  =4-18    36  28  .7. 


From  the  American  Nautical  Almanac  we  obtain,  for  the  true  places 
of  the  sun, 


o 

=   80°  24'  32".4, 

log/? 

=--  0.006774, 

O' 

=  110    55  51  .2, 

log/i' 

=  0.007101, 

G" 

=  141    33     2  .0, 

logi?" 

=  0.005405, 

the  longitudes  being  referred  to  the  mean  equinox  1858.0. 

When  the  ecliptic  is  the  fundamental  plane,  we  have,  neglecting 
the  sun's  latitude,  D  =^0,  and  we  must  write  ?.  and  ^9  in  place  of  a 
and  S,  and  O  in  place  of  A,  in  the  equations  which  have  been  derived 
for  the  equator  as  the  fundamental  plane.     Therefore,  we  have 


ff  cos(G  —  Q)  =  B"  cosiQ"  —  e    —R, 
gm\{G—0)  =  K'm\{Q"—Q); 

cos  4  =:  COS  /5  cos  (A  —  O),  cos  4"  =  cos  /S''  cos  (/" 

i?cos4=-6,  i?"  cos  4"  =  ft", 

i?  sin  4  =  if,  i?"  sin  4"  =  i?", 


O") 


from  which  to  find  G,  g,  h,  B,  b",  and  B",  all  of  which  remain 
unchanged  in  the  successive  trials  with  assumed  values  of  J.  Thus 
we  obtain 


G  =  201°  7'  57".4, 
log^r/  =  0.013500, 


logJB  =9.925092, 
log  B"  =  9.510309, 


b  =  +  0.568719, 
b"  =  4-  0.959342. 


Then   we  assume,   by   means   of  approximate   elements  already 

known, 

log  J  =  0.397800, 

and  from 

h'  cos  :'  cos  iH'  —  (?)  =  J  cos  /J  cos  (^  —  G)  4-  g, 
h'  cos  r  sin  {H'  —G)  =  d  cos  /?  sin  (A  —  G), 
h'  sin  Z'  =  'J  sin  /5, 

we  find  H',  ^',  and  h'.    These  give 
H'  =  153°  46'  20".5,         C  =  4-  7°  24'  16".4,         log  h'  =  0.487484. 


NUMERICAL   EXAMPLE. 

Next,  from 

cos  <p'  —  cos  :'  cos  /S"  cos  (A"  —  .ff')  +  sin  C  sin/5", 
/i'  cos  <p'  =  c,  A'  sin  <p'  =  C, 

wc  got 

log  (7:=  9.912519,  c=:  + 2.961673; 

and  from  _____^____ 

r  =  l/(J  — 6J^  + j^^ 
\vc  find 

logr  =  0.323446. 
Then  we  have 


2r'  2r' 


215 


j"  =  c±t/x'^  — CS 


(r  +  /')2 


—7-: ^-A*. 

l/r  +  r" 


from  which  to  find  J",  r',  and  x.     First,  by  means  of  the  approxi- 
mate elements,  we  assume 

log  J"  :=.  0.310000, 

which  gives  log  r"  =  0.053000,  and  hence  we  have 

ij=  0.3783,  log// =  0.002706,  log  x  r=  0.090511. 

With  this  value  of  x  wc  obtain  from  the  expression  for  J",  the 
lower  sign  being  used,  since  A"  is  less  than  c, 

log  J"  =  0.309717. 

Repeating  the  calculation  of  r", /-<,  and  x,  and  then  finding  A"  again, 

the  result  is 

log  J"  =  0.309647. 

Then,  by  means  of  the  formula  (67),  we  may  find  the  correct  value. 
Thus  we  have,  in  units  of  the  sixth  decimal  place, 

a  =  309717  —  310000  =  —  283,         a  =  309647  —  309717  =  -  -  7C, 

and  for  the  correction  to  the  last  result  for  log;  A"  we  have 


Therefore, 


^  =  -23. 

a — a 

log  A"  =  0.309624. 

By  means  of  this  value  we  get 

log  r"  =  0.052350,  log  x  =  0.090628, 


216  THEORETICAL  ASTRONOMY. 

and  tliis  value  of  X  gives,  finally, 

log  J"  =.  0.309623,  log  r"  =  0.052348. 

riie  lieliocentric  places  of  the  comet  are  now  found  from  the  e<jua- 
tions  (71)  and  (72),  writing  Jeos/3  and  J"c!os/3"  for  p  and  />", 
respectively.     Thus  we  obtain 

I  =  159°  43'  14".2,        b  =  +  10°  50'  14".0,        logr  =  0.323447, 
r^l44    17  47  .8,        6"  =  +  35    14  28  .7,        logr"  =  0.052347. 

The  agreement  of  these  results   for  r  and  r"  with   those  already 
obtained,  proves  the  accuracy  of  the  calculation.     8incc  the  helio- 
centric longitudes  are  dimiuishing,  the  motion  is  retrograde. 
Then  from  (74)  we  get 


and  from 


Q,  =  165°  17'  30".3, 


tan(7— Ji) 

tan  ?i  =^ . — -, 

cost 


tan  u 


i  =  63°6'32".5; 

tan(l"-Q) 


COSi 


we  obtain 


«--12°  10'  12".6, 


u"  =  40°  18'  51".2, 


the  values  of  —  ic  and  I  —  SI  being  in  the  same  quadrant  when  the 
motion  is  retrograde.     The   equation   (79)  gives  log  x  =  0.090630, 
which  agrees  with  the  value  already  found. 
Tiie  formulse  (81)  give 


v"  =  u" 


iu  =  129°  6'  46".3, 

and  hence  we  have 

y  =  u  —  w  ==  _  116°  56'  33'.7, 

from  which  we  get 

T==  1858  Sept.  29.4274. 

From  these  elements  we  find 
log  r'  =  0.212844,  v'  =.  -  107°  7'  34".0, 

and  from 


log  5  =  9.760326, 


(^  =  —  88°  47'  55".l, 


u 


21°  59'  12".3, 


we  get 


tan  {I'  —  R)  =;  —  cos  i  tan  u', 

tan  6'  =  —  tan  i  sin  {P  —  Q), 


r  =  154°  56'  33".4, 


b'  =  +  19°  30'  22".l. 


NUMERICAL   EXAMPLE.  217 

By  nioxins  of  these  and  the  values  of  O'  and  R',  wc  obtain 
-I'  =.  137°  39'  13".3,  /5'  =  +  12°  54'  45".3, 

ami  comparing  tlieso  results  with  observation,  we  have,  for  the  error 

of  die  middle  i)lace, 

C.-O. 


cos;j'a/.'  =  — 27".2, 


A/}'  =  —  23" 


.1. 


From  the  relative  positions  of  the  sun,  earth,  and  eomot  at  the 
time  /"  it  is  easily  seen  that,  in  order  to  diminish  these  residuals,  the 
gco('entri(!  distance  must  be  increased,  and  therefore  we  assume,  for 
a  .second  value  of  J, 

log  J  =  0.398500, 
from  which  we  derive 


^^'=153°  44'57".6, 

log  6'=-^  9.912587, 
log  J"  =  0.311054, 


:'  =  +  7°  24'26".l, 
lege  =  0.472115, 
log  /•"  =  0.054824, 


Then  we  find  the  heliocentric  places 


/  =:  159°  40'  3.3".8, 

b  =  +  10°  50'    8".6, 

r=144    17   12  .1, 

h"  =  +  35     8  37  .8, 

and  from  these. 

log  A' =  0.4H802(), 
log  r  =  0.324207, 
log  X  =  0.089922. 


logr   =0.324207, 
log  ,•"  =  0.054825, 


S^  =  165°  15'  41".l, 

« z=    12    10  30  .8, 

w  =  128    54  44  .4, 

T=  1858  Sept.  29.8245, 

v'  =  —  106°  55'  43".8, 

/'=       154    53  32  .3, 

/=      137    39  39  .7, 


■i  =  63°    2'49".2, 
«"  =  40    13  26  .0, 

log  q  =  9.763620, 

log/ =  0.214116, 

m'=  21°  59'  0".6, 
6'  =  +  19  29  31  .9, 
/5'  =  +12    55     2  .9. 


Therefore,  for  the  second  assumed  value  of  J,  we  have 


C.-O. 


cos/yAA'^.  — 1",5, 


A;S'  =  — 6".l. 


Since  these  residuals  are  very  small,  it  will  not  be  necessary  to 
make  a  third  assumption  in  regard  to  J,  but  we  may  at  once  <lerivo 
the  correction  to  be  applied  to  the  last  assumed  value  by  means  of 
the  equations  (109).     Thus  we  have 


a'  =  —  1.5, 


a 


=  —  27.2, 
5  log  J  = 


rf'=_6.1, 
0.000700, 


d"=-23.7, 


218 


THEORETICAL   ASTRONOMY. 


and,  expressing  ^  log  J  in  nnits  of  the  sixth  decimal  place,  these 
equations  give 

25.7.T  —  lOoO  =  0. 
17.0^  —  4270  =  0. 

Combining  these  according  to  the  method  of  least  squares,  we  got 

105  X  2.57  +  427  X  1.76 


«  = 


+  106. 


(2.57/  +  (1.76;'' 

Henee  the  corrected  vjilue  of  log  J  is 

log  J  =  0.398500  +  0.000106  =  0.398606. 

With  this  value  of  iog  J  the  final  elements  are  computed  as  already 
illustrated,  and  the  following  system  is  obtained: — 

T=  1858  Sept.  29.88617  Washington  mean  time. 

n^   86°  22'36".9  1    ,,        ^     .         __  ^ 
^ Ip.-    ^-  24   Qf    Mean  Lqumox  1 808.0. 

i  =    63      2  14  .2 
log  3  =  9.764142 

Motion  Retrograde, 

If  the  distinction  of  retrograde  motion  is  not  adopted,  and  we  regard 
i  as  susceptible  of  any  value  from  0°  to  180°,  we  shall  have 

t:  =  294°    8'  12".7, 
i  =  116    57  45  .8, 

the  other  elements  remaining  the  same. 

The  comparison  of  the  middle  place  with  these  final  elements 
gives  the  following  residuals: — ■ 

C  — O. 

cos  ,3  aA  =  +  0".2,  A/3  =  —  4".3. 

The.se  errors  are  so  small  that  the  orbit  indicated  bv  the  observed 
jjlaces  on  which  the  elements  are  based  differs  very  little  from  a 
parabola. 

When,  instead  of  a  single  place,  a  series  of  intermediate  places  is 
employed  to  correct  the  assumed  value  of  J,  it  is  best  to  adopt  tlie 
equator  as  the  fundamental  plane,  since  an  error  in  a  or  d  will  all'wt 
both  I  and  /5;  and,  besides,  incomplete  observations  may  also  be  used 


NUMERICAL   EXAM PI.E. 


219 


whon  the  fundainontal  plane  is  that  to  which  the  obsorvatioiis*  are 
diri'ctly  referred.  Further,  tlie  entire  grouj)  of  ofjuations  of  cf)!!- 
dition  for  the  determination  of  .r,  aecordinji  to  the  forintdie  (101)), 
must  be  eonibine<l  by  nudtiplying  each  equation  l>v  the  eoofheicnt  of 
a'  in  that  equation  and  taking  the  sum  of  all  the  equationti  thus 
formed  as  the  final  equation  from  which  to  find  ,v,  the  observations 
being  supposed  equally  good. 


220 


TUEORETICAL  ASTKOSOMY. 


CHAPTER   IV. 


DETEUMIXATION,  FROM  THRKK  COMI'I-ETK  OIlSEUVATloNS,  OF  THE  ELEMENTS  OF 
THE  Olilirr  OF  A  HEAVENLY  HODY,  INIXIDINO  THE  ECCKNTUICITY  OK  FoUM  VV 
THE   ( OXIC   SECTION. 

77.  TnK  f'ornuilio  which  have  thus  far  boon  dorivcd  for  tlio  dotcr- 
miiiation  of  the  olemeuts  of  the  orhit  of  a  hoavonly  body  by  means 
of  observed  plaeos,  do  not  suffice,  in  the  form  in  which  they  liave 
been  given,  to  determine  an  orbit  entirely  unknown,  except  in  tlie 
partlcidar  case  of  parabolic  motion,  for  whicii  one  of  the  elements 
becomes  known.  In  the  {general  case,  it  is  necessary  to  derive  at 
least  one  of  the  curtate  distances  without  makinjj  any  assumi)tion  as 
to  the  form  of  the  orbit,  after  which  the  others  may  be  found.  I>iit, 
preliminary  to  a  complete  investigation  of  the  elements  of  an  un- 
known orbit  by  means  of  three  complete  observations  of  the  body, 
it  is  necessary  to  provide  for  the  corrections  due  to  jjarallax  and  aber- 
ration, so  that  they  may  be  ai)plied  in  as  advantageous  a  manner  as 
possible. 

When  the  elements  are  entirely  unknown,  we  cannot  correct  tiic 
observed  places  directly  for  parallax  and  aberration,  since  both  of 
these  corrections  require  a  knowledge  of  the  distance  of  the  body 
from  the  earth.  But  in  the  case  of  the  aberration  we  may  either 
correct  the  time  of  observation  for  the  time  in  which  the  light  from 
the  body  reaches  the  earth,  or  we  may  consider  the  observed  place 
corrected  for  the  actual  aberration  due  to  the  eond)ined  motion  of  the 
earth  and  of  light  as  the  true  place  at  the  instant  when  the  light  left 
the  planet  or  comet,  but  as  seen  from  the  place  which  the  earth  occu- 
pies at  the  time  of  the  observation.  When  the  distance  is  unknown, 
the  latter  method  nuist  evidently  be  adopted,  according  to  which  we 
apply  to  the  observed  apparent  longitude  and  latitude  the  actual 
aberration  of  the  fixed  stars,  and  regard  this  place  as  corresponding 
to  the  time  of  observation  corrected  for  the  time  of  aberration,  to  be 
efTected  when  the  distances  shall  have  been  found,  but  using  for  the 
place  of  the  earth  that  corresponding  to  the  time  of  observation.  It 
will  appear,  therefore,  that  only  that  part  of  the  calculation  of  the 


DKTKIOIINATIOX    OF    AN    OIUUT. 


221 


(■IciiK'iits  wliicli  involves  the  titiics  of  oUscrvatioii  will  have  to  In-  rc- 
|Kat('tl  alter  the  eorresi»omliiifj:  (li>taiiees  of  the  hody  from  tlie  earth 
liiive  been  found.  First,  then,  by  nieans  of  the  a[»i)arent  obliijuity  of 
the  eeliptie,  the  observed  apparent  rij^ht  ascension  and  declination 
iiitist  !)(' converted  into  apparent  longitude  and  latitUv.e.  Let  /^  and 
,t.  respectively,  denote  the  observed  ajyparent  loiif^itiicie  and  latitude; 
iiiid  let  0,i  be  the  true  longitude  of  the  sun,  2',,  its  latitude,  and  A', 
it.>  distance  from  the  earth,  corresponding  to  the  time  of  observation. 
Then,  if  /  and  ,^  denote  the  longitude  and  latitude  of  the  planet  or 
coinct  corrected  for  the  actual  aberration  of  the  fixed  stars,  we  shall 
have 

/.  -/>.„  =  +  20".445  cos  (i  —  0„)  sec ,?  +  0"..^4:{  cos  (;.  -  281°)  sec  <?,   ^ 
,}_,j^=  —  20"A\'i  sin  (;.  —  QJ  sin  <3  —  0".o43  sin  (A  —  281°)  sin ,5^. 


In  computing  the  numerical  values  of  these  corrections,  it  will  be 
sutlicieutly  accurate  to  use  /„  and  ,9^  instead  of  /  and  ^9  in  the  second 
incmbers  of  these  e([uations,  and  the  last  terms  niay,  in  most  eases, 
he  neglected.  The  values  oi'  ?.  and  ,i  thus  derived  give  the  true  ])lace 
of  the  body  at  the  time  t  —  497'.78  J,  but  as  seen  from  the  place  of 
(he  earth  at  the  time  t. 

When  the  distiuice  of  the  planet  or  coniot  is  unknown,  it  is  impos- 
.*il)le  to  reduce  the  observed  place  to  the  centre  of  the  earth;  but  if 
we  conceive  a  line  to  be  drawn  from  the  body  through  the  true  place 
of  observation,  it  is  evident  that  were  an  observer  at  the  point  of 
intersection  of  this  line  with  the  plane  of  the  eclii)tic,  or  at  any  point 
in  the  line,  the  body  would  be  seen  in  the  same  direction  as  from  the 
iietiial  place  of  observation.  Hence,  instead  of  a})plying  any  correc- 
tion for  parallax  directly  to  the  observed  ai)parent  place,  we  may 
conceive  the  place  of  the  observer  to  be  changed  from  the  actual  [)lace 
to  this  point  of  intersection  with  the  ecliptic,  and,  therefore,  it  be- 
comes necessary  to  determine  the  position  of  this  point  by  means  of 
the  data  fiu'uished  by  observation. 

liCt  d(^  be  the  sidereal  time  corresponding  to  the  ti.nc  /„  of  obser- 
vation, (p'  the  geocentric  latitude  of  the  place  of  observation,  and  />y 
tiie  radius  of  the  earth  at  the  place  of  observation,  ex[)ressed  in  i)arts 
of  the  etpiatorial  radius  as  unity.  Then  ^„  is  the  right  ascension  and 
<f'  the  declination  of  the  zenith  at  the  time  t^.  Let  l^  and  b^  denote 
these  (piantities  converted  into  longitude  and  latitude,  or  the  longitude 
and  latitude  of  the  geocentric  zenith  at  the  time  <„.  The  rectangular 
co-ordinates  of  the  place  of  observation  referred  to  the  centre  of  the 


222 


TlI EOUETrCA  L    ASTRONOMY. 


earth  aiul  expressed  in  parts  of  the  mean  distunee  of  the  earth  from 
the  sun  us  the  unit,  will  be 

Vo  --  t'o  i^i"  '^0  ''O"* ''«  >*'»  fo> 
Zo  =  i>o  *^in  "o  >*>»  l>o' 

in  which;r„-=8".5711G. 

J^'t  J^  be  the  distance  of  the  phmet  or  eomet  from  the  true  place 
of  the  observer,  and  J,  its  distance  from  the  j)oint  in  the  ecliptic  to 
which  the  observation  is  to  be  reduced.  Then  will  the  co-ordinates 
of  the  place  of  observation,  referred  to  this  point  in  the  ecliptic,  be 

x,  =  (J,  —  J,)  cos  /?  cos^, 
y,  =:(J,  —  J„)coS(?sin-'., 
z,  =:(J,  —  Jjsiu/5, 

the  axis  of  x  being  directed  to  the  vernal  erpiinox.  Let  us  now 
designate  by  O  the  longitude  of  the  sun  as  seen  from  the  point  of 
reference  in  the  eeli})tic,  and  by  li  its  distance  from  this  point.  Theu 
will  the  heliocentric  co-ordinates  of  this  point  bo 

Z=  — A'cosO, 
r=  — A'siuQ, 
Z  =  0. 

The  heliocentric  co-ordinates  of  the  centre  of  the  earth  are 

Xo  =  —  Iio  COS  -0  cos  ©0, 
Yo--=^  —  Ii„  cos  2„ sin  O 0, 
Zq  =  —  lif,  sin  -,. 

But  the  heliocentric  co-ordinates  of  the  true  place  of  observation 

will  be 

X  +  x„  r+y„  Z-{-z„ 


or 


and,  consequently,  we  shall  have 


Zq  +  »0' 


i?  cos  O  —  (J,  —  Jj)  cos  /?  cos  A  =  jRg  cos  -„  cos  ©o  —  Po  sin  tt^  cos  b^  cos/j, 
a  sin  ©  —  (J,  —  Jq)  cos  ,3  sin  A  =  /^^  cos  -„  sin  ©,  —  />,  sin  -„  cos  b„  sin  /„, 
—  (^/  —  -Jo)  sbi  ,3         —  Bo  sin  2;  —  p^  sin  -„  sin  b^. 

If  we  suppose  the  axis  of  x  to  be  directed  to  the  point  whose  longi- 
tude is  ©Q,  these  become 


DETERMINATION   OF   AN   ORllIT. 


223 


E  cos  ( O  ~  0„)  —  ( J, 1„)  cos  ,5  cos  (A  —  0,)  = 

Ji„  cos  I'o  —  r„  ^i"  "o  t'"«  ^  cos  (/„  —  0„), 
li^m  (0  —  0„)  —  (J,  —  Jo)  co8,J  sin  (A  —  0„)  =^  (2) 

—  /'„  sill  T„  COS  Ao  sin  (■/„  —  0o), 
—  ( J,  —  Jj  sin  ,3  = /;„  sin  i;  — /v  sin  t:,,  sin />„, 


from  which  72  and  0  may  bo  (letcrniined.     Let  us  now  put 

(J, —  J„)  COS /?=:/) ; 


(3) 


then,  since  rr^,  1\,  and   0  —  0o  arc  small,  those  equations  may  be 

rwliiccd  to 

Ji  =  Dcos(X  —  0„)  —  7r„  />„  cos  b^  cos  (/„  —  ©„)  +  11^, 
EiQ  —  0o)  --  JJ sin  ( A  —  0 J  —  7r„ /;„ cos 6„ sin  (/„  —  0„), 
0  —  Z>  tan  ,J  —  Tu  ;r,^  sin  />„  +  ii'„  2;. 

Hence  we  shall  have,  if  rr„  and  -'„  arc  expressed  in  seconds  of  arc, 


5^sin6o-^ 

206264.8         " 


R  =i?o  +  -Dcos(>l  — 0„)  — -^' 


-aPoCOsbaCOsda—Qo. 


206264.8 


(4) 


0  =  Oo4- 


206264.8  D  sin  (X  —  0„)  —  -,  p„  cos  b„  sin  (/„  —  QJ 


from  which  we  may  derive  the  values  of  0  and  R  which  are  to  be 
u>'cd  throughout  the  calculation  of  the  elements  as  the  longitude  and 
distance  of  the  sun,  instead  of  the  corresponding  places  referred  to 
the  centre  of  the  earth.  The  point  of  reference  being  in  the  ]>lane 
of  the  ecliptic,  the  latitude  of  the  sun  as  seen  from  this  point  is  zero, 
which  simplifies  some  of  the  equations  of  the  problem,  since,  if  the 
observations  had  been  reduced  to  the  centre  of  the  earth,  the  sun's 
latitude  would  be  retained. 

We  may  remark  that  the  body  would  not  be  seen,  at  the  instant 
of  observation,  from  the  point  of  reference  in  the  direction  actually 
ohsorvcd,  but  at  a  time  different  from  ^„,  to  be  determined  by  the 
interval  which  is  required  for  the  light  to  pass  over  the  distance 
J,  —  Jy.  Consequently  we  ought  to  add  to  the  time  of  observation 
the  quantity 

(J,  —  J„)  497'.78  =  497'.78  D  sec  jS,  (5) 

which  is  called  the  reduction  of  the  time ;  but  unless  the  latitude  of 
the  body  should  be  very  small,  this  correction  will  be  insensible. 
The  value  of  k  derived  from  equations  (1)  and  the  longitude  © 


224 


THKOKKTirAI-    ASTIIONDMY. 


derived  from  (4)  slxtidd  he  rediieed  l)v  applying  the  eorreetioii  lor 
mitntioi)  to  tlie  mean  ecpiinox  of  the  date,  an<l  tlieit  l)oth  these  and 
the  hititncht  ^i  slionid  he  rechieed  hy  upplyinj^  the  eorreetion  tor  pro- 
cession to  tl»e  echptie  and  mean  ecpiinox  of  a  tixed  epoeli,  for  wliicli 
the  hejfiiininfjj  of  the  year  is  iisnally  ('h()sen. 

In  this  way  each  ohserved  ap{)arent  lonj^ituch!  and  hititnde  is  to  i)c 
corrected  for  the  al)erration  of  the  fixed  stars,  and  the  correspoiKhuj,' 
places  of  the  snn,  referred  to  the  point  in  which  the  line  drawn  from 
the  body  thron«>;h  the  jjlace  of  observation  on  the  earth's  surface  in- 
tersects the  plane  of  the  ecliptic,  are  derived  from  the  e»piations  (I). 
Then  the  places  of  the  stm  and  of  the  planet  or  comet  are  redii(0(l 
to  the  ecliptic;  and  mean  e(piinox  of  a  fixed  date,  and  the  results  tliii.s 
obtained,  toirether  with  the  times  of  observation,  furnish  the  data  lor 
the  determination  of  the  elements  of  the  orbit. 

When  the  distance  of  the  body  corres|)onding  to  cacli  of  the 
observations  shall  have  been  determined,  the  times  of  observation 
may  bo  corrected  for  the  time  of  aberration.  This  correction  is 
necessary,  since  the  adopted  places  of  the  body  arc  the  true  2)l!U'cs 
for  the  instant  when  the  light  was  emitted,  corresponding  respectively 
to  the  times  of  observation  diminished  by  the  time  of  aberration, 
hut  as  seen  from  the  places  of  the  earth  at  the  actual  times  of 
observation,  respectively. 

When  (9  -  0,  the  erpiations  (4)  cannot  be  applied,  and  when  tiie 
latitude  is  so  small  that  the  reduction  of  the  time  and  the  correction 
to  be  applied  to  the  place  of  the  sun  are  of  considerable  magnitutlc, 
it  will  be  advisable,  if  more  suitable  observations  are  not  available, 
to  neglect  the  correction  for  parallax  and  derive  thy  elements,  using 
the  uncorrected  [)laccs.  The  (li.sl:u>ces  of  the  body  from  the  earth 
which  may  then  be  derived,  will  (:nai>.<;  us  to  Jipi)ly  the  correction  for 
parallax  directly  to  the  observed  plsijos  of  the  body. 

When  the  approximate  distan'.'cs  of  the  body  from  the  earth  are 
already  known,  and  it  is  required  to  derive  new  elements  of  the 
orbit  from  given  observed  places  or  from  normal  places  derived  from 
many  observations,  the  observations  may  be  corrected  directly  for 
parallax,  and  the  times  corrected  for  the  time  of  aberration.  We 
shall  then  have  the  true  places  of  the  body  as  seen  from  the  centre 
of  the  earth,  and  if  tliese  places  are  adopted,  it  will  be  necessary,  for 
the  most  accurate  solution  possible,  to  retain  the  latitude  of  the  snn 
in  the  formula)  which  may  be  required.  But  since  some  of  these 
formulae  acquire  greater  simplicity  when  the  sun's  latitude  is  not 
introduced,  if,  in  this  case,  we  reduce  the  geocentric  places  to  the 


PKTKUMINATION    OF   AN   OlMUT. 


225 


piiiiit  ill  wliicli  a  |H-rp('i)(]irtiliii'  let  i'all  from  tlic  ctMitir  of  tlio  earth 
t(i  tlic  plaiK'  of  tli(>  ecliptic  cuts  that  phuio,  tiio  loiijritiide  of  the  suii 
will  reniain  uuchaiij;e<l,  the  latitiulo  will  l»o  zero,  and  the  (listanec  A* 
will  also  Im;  iiiichanj^ed,  since  the  {greatest  j^eoceiitric  latitude  of  the 
.still  (joes  not  exceed  1".  Then  the  loii^itiuh'  of  the  plaiief  or  C(tniet 
as  seen  t'rotii  this  point  in  the  ecli[)tic  will  he  the  same  as  seen  from 
till'  centre  of  the  earth,  and  if  J,  is  thi^  distance  of  the  body  from 
this  p(»int  of  reference,  and  ,9,  its  latitude  as  seen  from  this  point,  wo 
sliiill  have 

J,  cos  ,?,  ■:—  J  C09  ,?, 

J,  sin  (J,  =  J  sin  ,'i  —  lig  sin  1\, 

from  which  wo  easily  derive  the  corrcetion  ^9,  —  j3,  or  A;9,  to  he  applied 
to  the  j<;eoeentric  hititude. 


Thus,  we  find 


(6) 


2'„  hcing  ox]>ros.sed  in  seconds.  This  correction  having  been  applied 
to  tlie  geocentric  hititude,  the  latitude  of  the  sun  becomes 

.       1=0. 

The  correction  to  be  applied  to  the  time  of  observation  (already 
(liiiiinished  by  the  time  of  aberration)  due  to  the  distance  J,  -  J„ 
will  be  absolutely  insensible,  its  maximum  value  not  exceeding 
0'.0()2.  It  should  be  remarked  also  that  before  apj)lying  the  equa- 
tion (()),  the  latitude  -'„  should  be  reduced  to  the  fixed  ecliptic  which 
it  is  desired  to  adopt  for  the  definition  of  the  elements  which  deter- 
inino  the  position  of  the  plane  of  the  orbit. 

78.  When  these  })relimiiiary  corrections  have  been  applied  to  the 
(lata,  we  are  prepared  to  proceed  with  the  calculation  of  the  elements 
of  the  orbit,  the  necessary  formula)  for  which  we  shall  now  investi- 
gate. For  this  purpose,  let  us  resume  the  equations  (G).^ ;  and,  if  wc; 
multiply  the  first  of  these  equations  by  tan /9  sin/"  —  tan  ,9"  sin/, 
the  second  by  tan/9"  cos/  —  tan/9  cos/",  and  the  third  by  sin(^  —  /"), 
and  add  the  products,  we  shall  have 


(\=-)iR  (tan  ,5"  sin  (A  —  O)  —  tan  ,S  sin  (A"  —  ©)) 
-  r  (tan  ,5  sin  ( /"  —  A')  —  tan  ,3'  sin  (A"  —  A)  -\-  tan  /5"  sin  (A'  —  A)) 
~Ii'  (tan  ;5"  sin  (A  —  ©')  —  tan  ,3  sin  (A"  —  ©')) 
-}-  n"R"  (tan  ,i"  sin  (A  —  ©")  —  tan  ,3  sin  (A"  —  ©")). 


(7) 


It  should  be  observed  that  when  the  correction  for  parallax  is  applied 

15 


226 


TIIEOR  ETIO  A  L    A  ST  ItONOM  Y. 


to  the  rt\acc  (if  the  sun,  />'  is  the  projection,  on  the  plane  of  the 
celiptie,  of  the  distunee  of  the  body  from  tlie  point  of  reference  to 
whiclj  the  observation  lias  been  reduced. 

Let  us  now  desijunate  l)y  K  the  lon<fitude  of  the  ascenuliig  node, 
and  by  /  the  inclination  to  the  ecliptic,  of  a  great  circle  ])assnij2; 
through  the  iirst  and  third  observed  places  of  the  body,  and  we  have 


(8) 


tan/5  =  siu(A  -  7i')  tan /, 
tan  ,S"  =  sin  (A"  —  K)  tan  L 

Introducing  these  values  of  tan/9  and  tan/9"  into  the  equation  (7), 
since 

sin  (A  —  O)  sin  (A"  —  K)  —  sin  (A"  —q)  sin  (A  —  K)  = 

—  sin  (/."  —  .1)  sin  (G  —  K), 
sin  (;/  —  /.)  sin  (A"  —  K)  +  sin  (A"  -  X')  sin  U  -  A')  ::- 

4-  sin  (A"  —  A)  sin  (A'  —  K), 
sin  (A  —  O')  sin  (A"  —  K)  —  sin  (X"  ~  ©' )  sin  (A  —  K)  = 

_sin(A"  — A)sin(0'  — A'), 
sin  (A  —  O")  sin  (A"  —  K)  —  sin  (A"  -  ©")  sin  ( A  —  iJT)  = 

—  sin(A"  — Ajsin(0"  — 7v'), 

we  obtain,  by  dividing,  tl.rougli  by  sin  (/"  —  ?.)  tan  /, 

0  =  nB  sin  ( 0  —  A' )  +  /-'  (sin  (X'  —  K)  —  tan  fi'  cot  /) 

-  E'  sin  (O'  —  K)  +  },"Ii"  sin  (©"  —  A').  ^^' 

Let  ;9|,  denote  the  latitude  of  that  point  of  the  great  circle  passing 
through  the  first  s'.nd  third  places  whicfh  corresponds  to  the  longitude 
/',  then 

tan  ,^0  =  sin  (A'  —  A")  tan  /, 

and  the  coeifieient  oi  f)'  in  equation  (9)  becomes 

si»  (i%  --  n 

coS/SaC 
Therefore,  if  we  put 


we  shall  have 


COS,SoCOS;3'  tU!.  J 

^sinO?;;--./?,) 
»       cos  ;1  tan  J' 


/)'sce,5': 


A"  sin  ( 0'~  A")    ,      R  sin  ( ©  —  A') 

p  „, . 

Ofl  Wo 


(10) 


(11) 


This  fornnila  will  give  the  value  of  (>',  or  of  J',  when  the  values  of 
n  and  n"  have  !)een  determined,  since  a^  and  A'' are  derived  from  <ho 
data  furnislied  by  observation. 


DETERMINATION   OF    AN    ORiUT. 


227 


To  find  A' and  /,  we  obtain  from  ('({nations  (8)  l\v  a  tran.sfbrniatinn 
precisely  similar  to  that  by  which  tho  cqnations  [lo)^  woro  derived, 


tan  I  sm  il.  {).  -\-  a)  —  A  )  --  ^-        ,  -  — ^  sec  .',  ( /.  —  /), 
-  ■  -^2  cort  ti  cos  ,i 

tan  J  cos  (A  (A"  +  A)  _  A')  --=  '-'^4-—^,  cosec  .',  {)."  -  A). 
^-  ~      '  -*       2  cos/^  cos /J 


(12) 


"W'c  may  also  compntc  K  and  /  from  the  orinations  which  may  be 
derived  from  (74)3  and  (7()).,  by  makinj^  the  necessary  changes  in  the 
notatioi),  and  using  only  the  upper  sign,  since  /is  to  be  taken  always 
less  than  90°. 

Jiefore  proceeding  further  with  the  discussion  of  equation  (11),  let 
us  derive  expressions  for  p  and  ft"  in  terms  of  //,  the  signification  of 
!>  and  //',  when  the  corrcetio.is  for  parallax  arc  applied  to  the  places 
of  the  sun,  being  as  already  noticed  in  the  case  of  //. 

71).  If  we  multiply  the  first  of  equations  (6).j  by  sin  ©"  tan^i", 
the  second  by  — cos  O"  tan^?",  and  the  third  by  a'niU" —  0")j  ''"*^^ 
add  the  products,  we  get 

0-:^  H/'(tan;j'''sin(0''-A)---tan,?sin(0"— A"))— Hieva'.M}"sin(0"--0) 
-//  (tan  ii"  sin  ( ©"— A'  ^-tan  ,3'  „in  (©"—A"  ))^~U'  tan  [i"  sin  (.©"—©'), 

(13> 
which  may  be  written 

0=^»/'(tanr3sin(A"— ©")—tan,5"sin  (A—©"))— »ii!tan,j"  sin  (©"—©) 
+  />'(tan,J"sin(A'—  ©")  —  tan  ,:'osin  (A"-^  ©")) 

—  ^'(tan,j'— tan,J„;  sia(A"  ~  ©")  -j- A"  tan,?"  sin  C©"—  ©'). 

Introducing  In<-o  tliis  the  v.dues  of  tan  ^5,  tan^i",  and  tan,?u  in  terras 
of  /  and  A",  and  reducing,  the  result  is 

0  —  ?(/>  sin  (A"—  A )  sin  ( ©  "—  K)  —  uR  sin  (©"-©)  sin  (A"—  A') 
-^  />'sin  a"— I' )  sin  (  ©"—  K)  -  i>'a,  sec,j'  sin  (A"-  ©") 
+  A'  sin  (©"  —  ©  'j  sin  (A"  —  A^;. 


Tlioreibre  we  obtain 

''~n\  sin  (A""  —  A)  +  sin'CA"  — 


sin(A"— O") 


A)    sin  (©"—A')/ 
sin(A"— ii)    A^in (©"—©')— »7^sin (Q"~Q) 
sinCA"  — A)sin(u"  — A')    "   ' 


n 


But,  by  means  of  the  equations  (9)3,  we  derive 
R'  Hin(©"  —  ©')  —  nR  sin(©"  —Q)  =  (N—n)R  sin  (©"—  ©), 


228 


THL.)RETICAL   ASTRONOMY. 


and  the  preceding  equation  reduces  to 

_///sin(>l"-;/)  fl„  sees' 

P  —  —   znrrvr     rr  ~r 


sin  (/"—©") 


■X) 


sin  (A"—;.)    sin(0 


—  O")  \ 
"-K)j 


4-1  I  ^^\  ^^si"^Q"—  0)  sin  (/"  —  K) 
'^\  nj~  sin  (/''  —  A)  sin  CO"  —  A')    " 


(14) 


To  ol)tain  an  expression  for  p"  in  terms  of  />',  if  we  multiply  the 
first  of  equations  (6)3  by  sin  O  tan  /3,  the  second  by  —  cos  ©  tan  ,'i, 
and  the  third  by  sin  {?<  —  O),  and  add  the  products,  we  shall  have 

0=nV'(tan/3sin(r—O)— tan /S" sin  (A—©))— H"i2"  tan /3sin(O"—0) 
— o'(tan/Ssin  (/'—©)— tan/5' sin  (A— Q))+/i' tan, S  sin  (©'—©).  (lo) 

Introducing  the  values  of  tan/9,  tan/9',  and  tan/5"  in  tex-ms  of  ^ and 
/,  and  reducing  precisely  as  in  the  case  of  the  formula  already  fouD'j. 
for  /),  we  obtain 


^  ~«"\sin(A"— r 


sin  (A  —  ©) 
X)      sin(/."— A)'sin(©  —  A') 


o„  sec  ii 


) 


/         N"  \A^"sin(©"—  ©)sin(A  — A^X 
~^\  n"  J     sinU"  — A)sin(© 


(16) 


K) 


Let  us  now  put,  for  brevity. 


6  = 


is?  sin  (©  —  A) 


R'sm(O'-K) 


or 


,      i?"sin(©"-A)        . 

d  = ,  /  : 


sec  /5' 


sin  (/"—;.)' 


h  = 


/?i?"sin(©"-0) 


aoSinCA"-;.) 


M, 


sin  (A''  -//)         R"  sni  (;."  -  ©") 
I  J ' 


m: 


'  sin  [)."  —  I) 
sinU'— A) 


M,= 


sin  (;."  —  /) 
/j.sin(;."— AO 


-/ 


d 

) 

Asin(-l  — ©) 

h 

,,„ /(  sin(-i  - 

-A-) 

~                « 

(17) 


and  the  equations  (11),  (14),  and  (16)  become 


p'  sec  /5'  = 


c  +  ?i6  -f  n"d, 


P  ^- 


m: 


n 


+  i>/. 


+  M. 


(-f)' 
'(-?')■ 


(18) 


If  n  and  n"  are  known,  these  equations  w'll,  in  most  j^'ASCS 


sufficient  to  determine  (>,  ft',  and  (> 


be 


DETERMINATION   OF   AN   ORBIT. 


229 


80.  It  will  be  apparent,  from  a  consideration  of"  the  equations 
which  have  been  derived  for  p,  (»',  and  (>",  that  under  certain  circum- 
stances they  arc  inapplicable  in  the  form  in  which  they  have  been 
given,  and  that  in  seme  cases  they  become  indeterminate.  When  the 
groat  circle  passing  chrough  the  first  and  third  observed  places  of  the 
body  passes  also  tiirough  the  second  place,  we  have  a„  =  0,  and 
equation  (11)  reduces  to 

n'  R"m\{Q"  —  K)  +  nRsm{Q  —  K)  =  R'  siniQ' ~  K). 

If  the  ratio  of  n"  to  n  is  known,  this  equation  will  determine  the 
quantities  themselves,  and  from  these  the  radius-vector  r'  for  the 
middle  place  may  be  found.  But  if  the  great  circle  which  thus 
passes  through  the  three  observed  places  passes  also  through  the 
second  place  of  the  sun,  we  shall  have  K=  ©',  or  K^^  180°  +  ©', 
and  hence 


or 


7i"R"  sin (Q"  —  Q')  —  uR  sin  (Q'  - 

H^  _  R  sin  (Q^-Q) 
n       "ie"sin(0"  — O'}' 


o)  =  o, 


from  which  it  appears  that  the  solution  of  the  problem  is  in  this 
case  impossible. 

If  the  first  and  third  observed  places  coincide,  we  have  ^  =^  ?."  and 
^  -  ^",  and  each  term  of  equation  (7)  reduces  to  zero,  so  that  the 
pv  '/1cm  becomes  absolutely  indeterminate.  Consequently,  if  the 
i!ta  rt)'  nearly  such  as  tc  render  the  solution  impossible,  according 
t  u:(  ( onditions  of  these  two  cases  of  indetermination,  the  elements 
'v!'irh  may  be  derived  will  he  greatly  affected  by  errors  of  observa- 
tion. X<'  however,  /  is  equal  to  ?/'  and  /3"  differs  from  fi,  it  will  be 
possible  to  derive  f/,  and  hence  f>  and  ft" ;  but  t  le  formula;  Avliich 
luive  been  given  requii     some  modification  in  tl  is  particular  case. 


Thus,  when  k 


/' ,  we  i. 


K=X"  =  ^,  7  =  90°,  and  ;9,^-=90' 

0 


and  hence  «„,  as  determined  by  equation  (iO),  becomes  -.     Still,  in 

tliis  case  it  is  not  indeterminate,  since,  by  recurring  to  the  original 
equation  (9),  the  coefficient  of  f>',  which  is  — «„  sec/9',  gives 


o„  =  sin  /3'  cot  7 —  cos  /3'  sin  (/'  —  K), 
au'l  when  X  =  ^",  it  becomes  simply 

cfg  =  —  cos  ,5'  sin  (/' —  K). 


(19) 


230 


THEORETICAL   ASTRONOMY. 


Wlicnover,  therefore,  the  differeace  /"  —  X  is  very  small  compared 
with  the  motion  in  latitude,  a„  should  be  computed  by  means  of  the 
equation  (19)  or  by  means  of  the  expression  Avhich  is  obtained 
directly  from  the  coeflicient  of  p'  in  equation  (7). 

When  /  =-  /"  -=:  A',  the  values  of  .¥„  J//',  M.„  and  M,"  ca:-iot 
be  found  by  means  of  the  equations  (17);  but  if  we  use  the  original 
form  of  the  expressions  for  p  and  p"  in  terms  of  p',  as  given  by 
equations  (18)  and  (15),  without  introducing  the  auxiliary  angles, 
we  shall  have 


_(>'    tan,S';  : 
n    tau  (J  siu  ( 


-G^')  — tan,rsin(/'— 0") 
0") 

N 


n 


Hence 


tan  z'j"  sin  (;.  —  ©'') 

"^  \     '      11   )  tan  /?  sin  ( A"  —  ©'')  —  tan  /"J"  sin  (/. 
^n  /5^in  (/'  —  ©)  —  tan  ,5'  sin  (A  —  ©) 
tan  ,i  sinT^/'^^o7-~taii^i''  i*i^n (A^^^oT 

/ .  _  iV;  V irtaD/Ssin(Q^'-G) 

"'"  \         'n"  j  tan  /5  sin  U"  —  © )  —  tan  /j"  sin  (;. 

1/  —  ^"Zi.'"iA"  ~  Q'O  — tan;?''sin(/^—  Q") 


0")' 


-O) 


M, 


tan  /J  sin  (/"  —  0")  —  tan  ,5"  sin  (/ 
,f tan  /?  sin  (^'  —  ©)  —  tan  ,S'  sin  (k 


©")' 
-©)• 


iK 


iW  = 


tan  /5  sin  (A"  —  ©)  —  tan  ,'i"  sin  (A 

R  tan  /5"  sin  ( 0"  —  0) 

tan  /?  sin  (■  '  —  ©")  —  tan  ,f"  sin  (A  —  ©")' 

i^'^tan  /5  sin  ( ©"  —  ©) 

lanT^  sin  (A"-"-  ©j  —  tan /5"  sin  (A  —  ©)  ' 


(20) 


arc  the  expressions  for  3/,,  31/',  31.^,  and  31.,"  which  must  be  used 
when  /  -  -/"  or  when  X  is  veiy  nearly  equal  to  /";  and  then  p  and  //' 
will  be  obtained  from  equations  (18).  These  expressions  will  also  be 
used  when  A" —  /  =^  180°,  this  being  an  analogous  case. 

When  the  great  circle  passing  through  the  first  and  third  observed 
places  of  the  body  also  passes  through  the  first  or  the  third  ])lace  of 
the  sun,  the  last  two  of  the  equations  (18)  become  indeterminate,  and 
other  foi  .luke  must  be  derived.  If  wc  multiply  the  second  of  equa- 
tions (7)3  by  tan/5"  and  the  fourth  by  — sin  (A" —  ©'),  and  add  the 
products,  then  multiply  the  second  of  these  equations  by  tan  ,5  and 
the  fourth  by  — sin  (A —  ©'),  and  add,  and  finally  reduce  by  means 
of  the  relation 


wc  get 


NE  sin  (©'  -  0)  =  N"R"  sin  (©"  -  ©'), 


detj:umixation  of  ax  orbit. 


231 


(>'_    tan ,?"  sin  (// —  Q')  — tan  ,5'  sin  ( )!'  —  © ') 
n  '  tun  ,y'  sin  {?.  —  ©')  —  t"an  /J  sin  ( A"~—  Q ') 

/r  tan;/' sin  (O" 


Q') 


sin  ( /  —  ©')  —  tan  /J  sin  (A"  -  ©')' 
„^P^    tan ;/  sin  (A  —  ©')  -  tan  /?  sin  (/■'  —  ©') 
^'       n"  ■  tan  ;5"  sin  (I  —  ©0  —  tan  /J  sin  (/"  -  -  ©')  ^"  ^ 

"^  \  n,"  N"  }  tan  (J"  sin  ().  —  ©')  —  tun'^f  siu('<"—  ©')' 
Tlicsc  equations  ai'e  convenient  for  determining  ft  and  f/'  from  p' ; 
Ijiit  they  boeome  indetcriniiv:^  when  the  great  circle  passing  tlu'ougli 
the  extreme  phices  of  the  body  also  passes  tlirough  the  second  place 
of  the  sun.  Therefore  tliey  will  generally  be  inapplicable  for  the 
cases  in  which  the  equations  (18)  fail. 

If  we  eliminate  f/'  from  the  first  and  second  of  the  equations  (6)3 
wc  get 

0  =  )ip  sin  (/."  —  k)—nR  sin  (/."  —  ©)  —  p'  sin  (/."  —  A') 
+  R'  sin  (A"  —  ©')  -  H"Ii"  sin  (A"  —  ©"), 
from  which  we  derive 

//    sin  (A"-/') 
''■^.T-sinll'^-A)  ^22) 

uE  sin  (/"  —  ©)  -  -  .R'  sin  (A"  —  Q')  +  n"R"  sin  (>."  —  ©  ") 
"^  /i  sin  (A"  —  A) 

Eliminating  ^o  between  the  same  ecpiations,  the  resu^~  is 


^=,7' 


f/    sin(A'  — A) 


sin  (A"  — A) 

7iR  sin  (A 


©)  —  i?'  sin  (A  -  ©')  +  n"R"  sin  (A 


(23) 
©") 


n"  sin  (A"  —  A) 
These  formula}  will  enable  us  to  determine  f>  and  ,0"  frron  ,n'  in  the 
special  cases  in  which  the  equations  (18)  and  (21)  are  inapplicable; 
but,  since  they  do  not  involve  the  third  of  equations  (6)„  they  are 
not  so  well  adapted  to  a  complete  solution  of  the  jtroblem  as  the 
lOrmuUe  previously  given  whenever  tliese  may  be  applied. 

If  we  eliminate  successively  p"  and  ()  between  the  first  and  fourth 
of  the  equations  (7).j,  we  get 

_  p'    tan  ;5"  cos  (A'  ~  ©')  —  tan  ,3'  cos  (A"  —  ©') 


n     tan  /j"  cos  (A  —  ©')  —  tan  fi  cos  (A" 
I  ^}^^^    niicosC©'— ©)  — ii'- 


©') 


vi"/i"  cos  (©"-©') 


n  tan  /5"  cos  (A  —  © ')  —  tan  ,3  cos  (A"  —  ©') 

„      p'     tan  if  cos  (A  —  ©')  —  tan  ,9  cos  (A'—  © ') 


(24) 


n"    tan , J"  cos  ( A  —  © ')  —  tan  ,5  cos  ( A"—  © ' ) 

tan  /5    „ie  cos  ( © '  —  © )  —  i?'  +  «"i2"  cos  (©"—©') 


n" 


tan ;?"  cos  (A  —  ©')  —  tan  /?  cos  (A"  —  ©') 


232 


TIIEORETU  AL   ASTRONOMY. 


Mliic'li  may  also  be  used  to  dctorniine  />  and  ft"  when  the  equationa 
(18)  and  (21)  cannot  be  applied.  When  the  motion  in  latitude  is 
greater  than  in  longitude,  the.se  equations  are  to  be  preferred  in!«tead 
of  (22)  and  (23.) 

81.  It  would  appear  at  first,  without  examining  the  quantities  in- 
volved in  the  formula  for  (>',  that  the  equations  (26).,  will  enable  us 
to  find  11  and  n"  by  successive  approximations,  assuming  first  that 


n  =  ,-> 

T 


n 


r" 


and  from  the  resulting  value  of  p'  determining  /•',  and  then  carrying 
the  approximation  to  the  values  of  n  and  n"  one  step  farther,  so  as 
to  include  terms  of  the  second  order  with  reference  to  the  intervals 
of  time  between  the  observations.  But  if  we  consider  the  equation 
(10),  Ave  observe  that  a„  is  a  very  small  quantity  depending  on  the 
difference  ^'  —  /9„,  and  therefore  on  the  deviation  of  the  observed 
path  of  the  body  from  the  arc  of  a  great  circle,  and,  as  this  a^jpears 
in  the  denominator  of  terms  containing  n  and  n"  in  the  equation 
(11),  it  becomes  necessary  to  determine  to  what  degree  cI  approxi- 
mation these  quantities  must  be  known  in  order  that  the  resulting 
value  of  (»'  may  not  be  greatly  in  error. 

To  determine  the  relation  of  Oy  to  the  intervals  of  time  between 
the  observations,  we  Jiave,  from  the  coefficient  of  p'  in  equation  (7), 


Oo  sec  ;i'  =  tan  ,3  sin  (A"  —  k')  —  tan  /S'  sin  (A"  —  A)  -[-  tau/S"  sin  (A'  —  /). 

We  may  put 

tan  ,3   =  tan  ;/  —  At"  -\-  Br'"'  —  ...., 

tan  ,5"  =  tan  /5'  -\- At -\- Bt' -\- . . . . , 

and  hence  we  have 


flo  sec  /5'  =  (sin  (/."  —  A')  —  sin  (A"  —A)  +  sin  (A'  —  A))  tan  /5' 
-f  (r  sin  {X'—?.)  —  r"  sin  (A"— A'))  ^+(r^  sin  (A'— A)+r"^  sin  {a"— a'))  B-\-.  ., 

which  is  easily  transformed  into 

cfo  sec  /5'  =  4  sin  i  (A'  —  A)  sin  ^  (A"  —  A')  sin  A  (A"—  A)  tan  ,5'      (2o) 
-f-  (r  sin  {k'—k)~T"  sin  (A"— A')M+(r"  sin  a'—).)+T"-'  sin  (A"— A'))J5-f. ... 

If  we  suppose  the  intervals  to  be  small,  we  may  also  put 

sinV(A"  — A)  =  ^(A"  — A), 
and 

sin  (A"  —  k)-=  A"  —  A,  sin  (A'  _  A)  =  A'  —  A. 


DETERMINATION   OF   AN   ORBIT. 

Furtlicr,  we  may  put 

k  =  A' -  ylV  +  B'r'"  —  .  .., 

A"  =  A' 4- yl'r  +  5V  + 


233 


Sulistituting  tlicse  values  in  the  equation  (25),  neglecting  terms  of 
tlie  fourth  order  with  respect  to  r,  and  reducing,  we  get 

a,  ^  tt't"  (4.4"  tan  ,5'  +  A'B  —  AB')  ccs  ;5'. 

It  appears,  therefore,  that  a^  is  at  least  of  the  third  order  Avitii 
reforonce  to  the  intervals  of  time  between  the  observations,  and  that 
an  error  of  the  second  order  in  the  assumed  values  of  n  and  n"  may 
produce  an  error  of  the  order  zero  in  the  value  of  {>'  as  derived  from 
0(Hi!ition  (11)  even  under  the  most  favorable  circumstances.  Hence, 
in  general,  we  cannot  adopt  the  values 


?i  =  -r> 


II -T) 

T 


omitting  terms  of  the  second  order,  without  affecting  the  resulting 
value  of  f/  to  such  an  extent  that  it  cannot  be  regarded  even  as  an 
apj)ro.\imation  to  the  true  value ;  and  terms  of  at  least  the  second 
order  must  be  included  in  the  first  assumed  values  of  a  and  n". 
The  equation  (28)3  gives 


(26) 


dr 


omitting  the  term  multiplied  by    ,-.  which  term  is  of  the  third  order 
with  respect  to  the  times ;  and  hence  in  this  value  of  — ,  only  terms 

.'at  least  the  fourth  order  are  neglected.     Again,  from  the  equations 

(26)3  we  derive,  since  r'  =  r  +  r", 


n  +  n"  =  1  +  -^ji, 


% 


,'3' 


(27) 


in  which  only  terras  of  the  fourth  order  have  been  neglected.     Now 
the  first  of  equations  (18)  may  be  written : 


/,'sec/5'  =  (?i  +  n") 


6  + 


1+^ 


(28) 


in  which,  if  we  introduce  the  values  of  —  and  n  +  w"  as  given  by 
(26)  and  (27),  only  terms  of  the  fourth  order  with  respect  to  the 


234 


THEORETICAL   ASTRONOMY. 


times  will  be  noffjec^ted,  and  consofjuently  the  rcsultiii}^  value  of  r/ 
will  1)0  att'cctcd  with  only  an  error  of  the  ^eeond  order  when  <i^,  i.s  of 
the  third  onler.  Further,  if  the  intervals  between  the  observations 
are  not  very  une<|Ual,  r*  —  r"'  will  be  a  (quantity  of  an  order  superior 
to  r",  and  when  these  intervals  arc  equal,  we  have,  to  terms  ot"  the 
fourth  order, 


The  equation  (27)  gives 


n 
u 


Hence,  if  we  put 


2/n«+«"-l)  =  -r". 


n 


(29) 


Q  =  2>'"  (n  +  n"  —  1), 
we  may  adopt,  for  a  first  approximation  to  the  value  of  />', 


P  = 


(;]0) 


and  f/  will  be  affected  with  an  error  of  the  first  order  when  the  in- 
tervals are  unequal ;  but  of  the  second  order  only  when  the  intervals 
are  equal.  It  is  evident,  therefore,  that,  in  the  selection  of  the 
observations  for  the  determination  of  an  unknown  ori)it,  the  in- 
tci'vals  should  be  as  nearly  equal  as  possible,  since  the  nearer  tliov 
approach  to  equality  the  nearer  the  trutli  will  be  the  first  assumed 
values  of  i*  and  Q,  thus  facilitating  the  successive  approximations; 
and  when  «„  is  a  very  small  quantity,  the  equality  of  the  intervals 
is  of  the  greatest  importance. 
From  the  equations  (29)  we  get 


71   -'■- 


1+P\    ^2?'='/' 
n"  =  nP; 

and  introducing  P  and  Q  in  (28),  there  results 

^Pd 

+  P' 


(31^ 


,'sec.S'  =  (l  +  ^)^ 


(32) 


This  equation  involves  both  f>'  and  r'  as  unknown  quantities,  but 
by  means  of  another  equation  between  these  quantities  f/  may  ho 
eliminated,  thus  giving  a  single  equation  from  which  r'  may  be 
found,  after  which  p'  may  also  be  determined. 


Dtn'KRMIXATIOX    OF   AX    ORBIT. 


235 


82.  Let  ^'  reprc'si'iit  thu  aiiijlo  at  the  earth  between  the  sun  and 
])lan(>t  or  co'net  at  the  sceunJ  ub.servation,  and  we  shall  have,  from 
the  eqiiations  (OS),, 


tan ,'/ 

tan  It'  --• '.—  ;-, 


Hin(/.'—  O')' 

,      tauU'— ©') 
tan  4  = ; , 

COSU' 
COS  4'  —  COS  (i  COS  (//  —  ©'), 


(33) 


l)v  means*  of  which  wc  may  determine  ^',  whicli  cannot  exceed  180°. 

Since  COS,*'  is  always  positive,  cosa^'  and  cos(/'— ^O')  must  have  the 

same  sij»'n. 

Wc  also  have 

r'^  =  J'^  +  i2'»  —  2  J'i?'  cos  4',  ^ 

which  may  be  put  in  the  form 

r'^  =  (/>'  sec  ,5'  —  E  cos  ijy  +  E-  siu'^  4', 
from  which  we  ffet 


//  sec /5'  =  E  cos ^'  ±iV r"  —  E' sin^ 4'.  (3t) 

Substituting  for  p'  sqc^3'  its  value  given  by  equation  (32),  we  have     . 

{i-^§,y'^^-c^Ecos^'±V7^=nr^i^. 

For  brevity,  let  us  put 


'■"  ~  '1  +  P' 

Cg  C  =  Kq, 

and  we  shall  have 

k„  —  4  =  E  cos  4'  =h  1/7^^— ie'^sin'H'. 


(35) 


(36) 


AVhen  the  values  of  P  and  Q  have  been  found,  this  equation  will 
give  the  value  of  r'  in  terms  of  quantities  derived  directly  from  the 
(lata  furnished  by  observation.  We  shall  now  represent  by  2'  the 
angle  at  the  planet  between  the  sun  and  earth  at  the  time  of  the 
second  observation,  and  we  shall  have 


E  sin  V 
sin  z' 


(37) 


236 


THEORETICAL  ASTRONOMY. 


Subsstitutiii};  this  value  of  /•',  in  the  precoding  equation,  there  results 


and  if  we  put 


(/'o  ~  K  cos  4.')  siu  z'  ^  y^"  sin  4'  cos  z'  =  A  .  3-7, 


ij„  sin  r  =  J{'  m\  V, 

5;„  cos  C  =:  /•„  —  W  COS  4', 

,o/rsin»V' 


(38) 
(39) 


JH„ 


the  condition  being  imposed  that  wip  shall  always  be  positive,  we 

have,  finally, 

sin  {z  =h  »)  =  (Ho  sin*/.  (40) 

In  order  that  m^  may  be  ])ositive,  the  quadrant  in  which  ^  is  taken 
must  be  such  that  -y^  shall  have  the  same  sign  as  /„,  sinca  sin  \^'  is 
always  positive. 

From  equation  (37)  it  appears  that  sin  z'  must  always  be  positive, 
or3'<180°;  and  further,  in  the  plane  triangle  formed  by  joining 
the  actual  places  of  the  earth,  sun,  and  planet  or  comet  corresponding 
to  the  middle  observation,  we  have 


A'^ 


Therefore, 


/_sm  iz'  +  40  _  R'  sin  iz' -\- ^') 
sin  4.'  sin  z' 

,      i2'sin(2'+4') 


sm2 


cos  /5', 


(41) 


and,  since  p'  is  always  positive,  it  follows  that  sin  (2'  +  1^')  must  be 
positive,  or  that  z'  cannot  exceed  180°  —  '^'. 

When  the  planet  or  comet  at  the  time  of  the  middle  observation  is 
both  in  the  node  and  in  opposition  or  conjunction  with  the  sun,  we 
shall  have /5'  =  0,  i|/' =  180°  when  the  body  is  in  opposition,  and 
'^'  =-  0°  when  it  is  in  conjunction.  Consequently,  it  becomes  impos- 
sible to  determine  r'  by  means  of  the  angle  2';  but  in  this  case  the 
equation  (36)  gives 


k 


I 


0      ps 


R'  +  r', 


when  the  body  is  in  opposition,  the  lower  sign  being  excluded  by  the 
condition  that  the  value  of  the  first  member  of  the  equation  must  be 
positive,  and  for  ^'  =  0, 

I 


I '0 

'^0      ^ 


R' 


the  upper  sign  being  used  when  the  sun  is  between  the  earth  and  the 


piyrKioirxATioN  of  an  ounrr. 


237 


pliinot,  and  the  lower  sij^n  when  the  i)l!iiiet  is  between  the  eartli  and 
tlic  snn.  It  is  hardly  necessary  to  remsirk  that  tlie  case  of  an  obser- 
vation at  tlie  superior  eonjnnetion  when  /9'  =  0,  is  i)hysieully  impos- 
,<il)le.     The  value  of  /'  may  be  i!>und  from  these  eijuations  by  trial ; 

and  then  we  shall  have 

p'  =  ,•'  -  W 

when  the  body  is  in  opposition,  and 

,,'  =  R'  —  r' 

when  it  is  in  inferior  conjunction  with  the  sum. 

For  the  case  in  which  the  great  circle  passing  through  the  extreme 
observed  places  of  the  body  passes  also  through  tiie  middle  place, 
wliicii  gives  o^ --  0,  let  us  divide  equation  (32)  through  by  c,  and  we 
have 

\     ^  2/'»  /  1  +  P 


/>'  sec  /5' 


The  equations  (17)  give 

^>_J?siu(0  —K) 
~c'~  A"sinCO'— isT)' 


and  if  we  put 


we  shall  have 


'L  +  p'l 

c     '         c 


d 

c 


t-'ni 


i2"sin(0"j 
ie'"sin(0'- 


-K) 


1+P 


since  c  =  co  when  a,,  =  0.     Hence  we  derive 


(-42) 


But  when  the  great  circle  passing  through  the  three  observed  places 
passes  also  through  the  second  place  of  the  sun,  both  c  and  C„  be- 
come indeterminate,  and  thus  the  solution  of  the  problem,  with  the 
given  data,  becomes  impossible. 

83.  The  equation  (40)  must  give  four  roots  corresponding  to  each 
sign,  respectively ;  but  it  may  be  shown  that  of  these  eight  roots  at 
least  four  will,  in  every  case,  be  imaginary.  Thus,  the  equation  may 
be  written 


?n„  sin*  z'  ■ 


sm  z  cos  ;  = 


cos  z  sm 


2.'J8 


TIIF/HJr.TK  A  r,    A.vrnoNOM  V. 


and,  1)V  squaring;  iiixl  ri'ilucinjrj  this  becomes 


»/( 'siii''2' 


2)ii„  cos  X  sin*  z'  -\-  m\^  z'  —  sin'''  C  —  0. 


WluMi  ^  is  witliin  the  limits  — 90°  and  -f  90°,  cos^  will  ho  ])ositi\v, 
and,  «/„  hcinji'  always  |)()sitivo,  it  appears  from  tlie  aljfehraie  si^ns  i,[' 
the  terms  of  the  eijuation,  aeeordini;-  to  the  theory  of  (>(|nation-;,  th;it 
in  this  ease;  there  cannot  he  more  than  tl)nr  real  rc»ots,  of  which  thnv 
will  l)(!  positive  and  (»ne  neu,ative.  When  ^  exceeds  the  limits  -  \HP 
and  I  90°,  cos  ^  will  ho  ne!j;ative,  and  hence,  in  this  ease  als(»,  there 
cannot  he  more  than  fonr  real  roots,  of  which  ono  will  he  [xtsitivo 
and  thre(!  nef:;ativc.  Further,  since  sin"  ;J^  is  real  and  ])ositive,  tlurc 
must  he  at  least  two  real  roots — one  [)ositiv(!  and  the  other  nej^ative 
— whether  cos  ^  he  ne}j;ative  or  i)ositive. 

We  may  also  remark  that,  in  finding  the  roots  of  the  e(piation  ( 10), 
it  will  only  ho  necessary  to  solve  the  cfjuation 


sin  (z'  —  C)  =  ■»»„  sin*  /, 


(4:!) 


since  the  lower  sign  in  (40)  follows  directly  from  this  hy  s;d)stittiti- 
180°  —  z'  in  j)lace  of  :;';  and  hence  the  roots  derived  from  tiiis  w 
comi)rise  all  the  real   roots   belonging  to  the  general  form  of  the 
equation. 

The  observed  places  of  the  heavenly  body  oidy  give  the  diroctina 
in  s|)ace  of  right  lines  passing  through  the  places  of  the  earth  iiiul 
the  corresponding  places  of  the  body,  and  any  three  points,  one  in 
each  of  these  lines,  which  are  situated  in  a  plane  passing  through  tlic 
centre  of  the  sun,  and  which  are  at  such  distances  as  to  fullil  the 
condition  that  the  aroal  veUx'ity  shall  be  constant,  according  to  tlio 
relation  expressed  by  the  equation  (30),,  must  satisfy  the  analytic;!! 
conditions  of  the  problem.  It  is  evident  that  the  three  places  of  tlie 
earth  may  satisfy  these  conditions ;  and  hence  there  may  be  one  root 
of  equation  (43)  which  will  correspond  to  the  orbit  of  the  earth,  or 
give 

Further,  it  follows  from  the  equation  (37)  that  this  root  must  be 

s'  =  180°  —  4' ; 

and  such  Avould  be  strictly  the  ease  if,  instead  of  the  assumed  values 
of  P  and  Q,  their  exact  values  for  tlie  orbit  of  the  earth  were  adopted, 
and  if  tiio  observations  were  referred  directly  to  the  centre  of  the 
earth,  in  the  correction  for  parallax,  neglecting  also  the  perturbations 
in  the  motion  of  the  earth. 


DKTKRMINATION    OF    AN    ORIJIT. 

In  the  cn.se  of  the  enrth, 


239 


n 


=  N  = 


/?'W"  sin  (©"-©) 


l{R"m\(Q"'-Q)' 
anil  iht'  coinplctt}  vulucs  ol'  P  ami  Q  beeomo 
/.'/."siii(0' -  ©) 

K{-Ui  y  J{Ii"mi(Q"~Q) 


''-^)' 


and  !«iiK'e  the  appi'oxiiniitc  values 


T 


Q  =-  rr" 


(litfcr  l)ut  little  from  these,  as  will  ai)p(>ar  from  the  equations  (27)^, 
tlKi'f  will  be  one  root  of  o(juation  (43)  whieh  gives  z'  nearly  eijual 
to  1S0°  —  a]/'.  This  root,  however,  cannot  satisly  the  physical  con- 
(litiniis  of  the  problem,  wliich  will  rc(pilre  that  the  rays  of  light  in 
coming  from  the  planet  or  comet  to  the  earth  shall  proceed  from 
points  whieh  are  at  a  considerable  distance  from  the  eye  of  the 
ol)scrver.  Further,  the  negative  values  of  sin  ;:'  arc  (excluded  by  the 
nature  of  the  problem,  since  r'  must  be  positive,  or  z'  <  180°  ;  and 
of  the  three  positive  roots  which  may  result  from  equation  (43),  that 
being  excluded  which  gives  s'  very  nearly  equal  to  180°  —  ^',  there 
will  remain  two,  of  which  one  will  be  excluded  if  it  gives  z'  greater 
tiian  180°  — 1|/',  and  the  remaining  one  Avill  be  that  which  belongs 
to  the  orbit  of  the  planet  or  comet.  It  may  happen,  however,  that 
neither  of  these  two  roots  is  greater  than  180°  —  -y^',  in  which  case 
both  will  sausfy  the  physical  conditions  of  the  problem,  and  hence 
the  observations  will  be  satisfied  by  two  wholly  different  systems  of 
elements.  It  will  then  be  necessary  to  compare  the  elements  com- 
puted  from  each  of  the  two  values  of  z'  with  other  observations  in 
order  to  decide  which  actually  belongs  to  the  body  observed. 

In  the  other  case,  in  which  cos  !^  is  negative,  the  negative  roots 
being  excluded  by  the  condition  that  r'  is  positive,  the  positive  root 
must  in  most  cases  belong  to  the  orbit  of  the  earth,  and  the  three 
observations  do  not  then  belong  to  the  sanie  body.  However,  in  the 
case  of  the  orbit  of  a  comet,  when  the  eccentricity  is  large,  and  the 
intervals  between  the  observations  are  of  considerable  magnitude,  if 


240 


TIIKOIU'/riCAT.    ASTRONO>fY, 


the  approximate  viiluos  of  P  and  Q  are  computed  directly,  by  moam 
of  ap])ro.\inuite  elemeiu.s  already  known,  from  the  equations 


rr'  )i\n{u'—n)  ♦ 

/>•"  sin  («"—«')' 
n  _  9  '3 /  rr'»in(u'—u)  +  r'r"  sin(u"— 7t')         \ 
V-^'-  \  rr"  sfn  (.«"- «)  V' 


(-i 


44) 


it  may  occur  that  eos  ^  is  negative,  and  the  positive  root  will  actually 
belontr  to  the  orbit  of  the  eomct.  The  condition  that  one  value  of 
s'  shall  be  very  nearly  e([ual  to  180°  — i^',  requires  that  the  adopted 
values  of  7^  and  Q  shall  differ  but  little  from  those  derived  directly 
from  the  places  of  the  earth  ;  and  in  the  case  of  orbits  of  small 
eccentricity  this  condition  will  always  be  fulfilled,  unless  the  intervals 
between  the  observations  and  the  distance  of  the  planet  from  the  sun 
are  both  very  great.  But  if  the  eccentricity  is  large,  the  difference 
may  be  such  that  no  root  Avill  correspond  to  the  orbit  of  the  earth. 

84.  We  may  find  an  expression  for  the  limiting  values  of  ?/!,,  and 
^,  within  which  ecpiation  (43)  has  four  real  I'oots,  and  beyond  which 
there  are  only  two,  one  positi\e  and  one  negative.  This  chaniie  iu 
tlie  number  of  real  roots  Avill  take  place  A\hen  there  are  two  ccpial 
roots,  and,  consequently,  if  we  proceed  under  the  supposition  that 
equation  (43)  has  two  c(pial  roots,  and  find  the  values  of  >/)(,  and  ^ 
which  Avill  accord  with  this  supposition,  wo  may  determine  the  limits 
recpiired. 

Differentiating  equation  (43)  with  respect  to  z',  we  get 


cos  (z'  —  X)  ---  4>«,  sin  V  cos  z' ; 


and,  in  the  case  of  equal  roots,  the  value  of 
must  also  satisfy  the  original  equation 


as  derived  from  tliis 


sin  (z'  —  :) 


^oSinV. 


To  find  the  values  of  vIq  and  ^  which  will  fulfil  this  condition,  if  we 
eliminate  m^  between  these  equations,  we  have 

sin  z*  cos  {z'  —  0  =  4  cos  z'  siu  (2'  —  i'), 

from  which  we  easily  find 

sin  (2/  —  C)  =  «  sin  C.  (46) 

This  gives  the  value  of  ^  in  terms  of  z'  for  which  equation  (43)  has 


DETERMINATION   OF   AN   ORBIT. 


241 


equal  roots,  and  at  whicli  it  cea.ses  to  have  four  real  roots.     To  find 
the  corresponding  expression  for  ?<*„,  we  have 


m„ 


sin  (z'  —  C)        cos  (/  —  C) 


sin  *z' 


4sinV  cof  z' 


in  which  we  must  use  the  value  of  ^  given  by  the  preceding  equation. 

Now,  f'nce  sin  (2s'  —  i^)  must  be  within  the  limits  — 1  and  H   1,  the 

limiting  values  of  sini^  will  be  +  ?  iuid  — §,  or  ^  must  be  within  the 

limits  +  30°  52'.2  and  —  86°  52'.2,  or  143°  7'.8  and  21G°  52'.2.     If 

(^  is  not  contained  within  these  limits,  the  equation  camiot  have  equal 

roots,  whatever  may  be  the  value  of  m^,  and  hence  there  can  only  be 

two  real  roots,  of  which  one  will  be  positive  and  one  negative.     If 

for  a  given  value  of  ^  ■  vc  compute  z'  from  equation  (45),  and  call 

this  z^\  or 

sin  (22/  —  »)  =  3*  sin  C, 

we  may  find  the  limits  of  iiic  values  of  77)^,  within  which  equation 
(43)  has  four  real  roots.     The  equation  for  zj  will  be  satisfied  by 

the  values 

2V-:,  180° -(22,-:); 

and  hence  there  will  be  two  values  of  ?»„,  which  we  will  denote  by 
Hii  iMid  JHj,  for  which,  with  a  givei;  value  of  !^,  equation  (43)  will 
have  equal  roots.     Thus  we  shall  have 


'"i  = 


sin  (2,,'  —  O 
sin*2g'      ' 


and,  putting  in  this  equation  180°  —  (2:;/  —  c^)  instead  of  2z^/  —  i^,  or 
90°  —  [zq  —  ^)  in  place  of  zj, 

cos  z^' 

ll  follows,  therefore,  that  for  any  given  value  -  .'  ^,  if  »(„  is  not 
within  the  limits  assigned  by  the  values  of  »(,  and  m.,,  equation  (43) 
will  oidy  have  two  real  roots,  one  positive  and  one  negative,  of 
which  the  latter  is  excluded  by  the  nature  of  the  problem,  and  the 
tbnnor  may  belong  to  tlie  orbit  of  the  earth.  But  if  P  arid  (^  ditler 
so  nuich  from  their  values  in  the  case  of  tlie  orbit  of  the  earth  that 
z'  is  not  very  nearly  ecpial  to  180°  —  4''>  ^''"^  positive  root,  when  ^ 
exceeds  the  limits  +  36°  52'.2  and  -  36°  o2'.2,  may  actually  satisfy 
the  eonditioris  of  the  problem,  and  belong  to  the  orbit  of  the  body 
observed. 

16 


242 


THEORETICAL  ASTBONOMY. 


AVlion  C  is  within  the  limits  143°  7'.8  and  216°  52'.2,  there  ^vill 
be  four  real  roots,  one  positive  and  three  negative,  if  m^  is  within  the 
limits  «i,  and  m.^ ;  but,  if  riiQ  surjrasscs  these  limits,  there  will  be  only 
two  real  roots. 

Table  XII.  contains  for  values  of  C  from  —  36°  52'.2  to  +  36°  52'.2 
the  values  of  my  and  r,i.,,  and  also  the  values  of  the  four  real  roots 
corresponding  respectively  to  ?/ij  and  wij- 

In  every  case  in  which  equation  (43)  has  three  positive  roots  and 
one  negative  root,  the  value  of  m^  must  be  Avithin  the  limits  indicated 
by  m,  and  m^,  and  the  values  of  z'  will  be  within  the  limits  indicated 
by  the  quantities  corresponding  to  mj  and  nij  for  each  root,  which 
Ave  designate  respectively  by  2/,  z./,  z^',  and  z/.  The  table  will  show, 
from  the  given  values  of  hIq  and  180° — 4'',  whether  the  probleia 
admits  of  two  distinct  solutions,  since,  excluding  the  value  of  z', 
which  is  nearly  equal  to  180°  — -x//'?  ^^^  corresponds  to  the  orbit  of 
the  earth,  and  also  that  which  exceeds  180°,  it  will  appear  at  once 
whether  one  or  both  of  the  remaining  two  values  of  z'  will  satisfy 
the  condition  that  z'  shall  be  less  than  180°  —  1]/'.  The  table  will 
also  indicate  an  aj)proxiraate  value  of  z',  by  means  of  which  the 
equation  (43)  may  be  solved  by  a  few  trials. 

For  the  root  of  the  equation  (43)  which  corresponds  to  the  orbit 
of  the  earth,  Ave  have  />'  =  0,  and  henco  from  (36)  Ave  derive 


k 


Jo 


Substituting  this  value  for  k^  in  the  general  equation  (32),  Ave  have 

and,  since  p'  must  be  positive,  the  algebraic  sign  of  the  numerical 
A'alue  of  /„  Avill  indicate  Avhethcr  r'  is  greater  or  less  than  li'.  It  is 
easily  seen,  froni  the  forniulre  for  /„,  b,  d,  &c.,  that  in  the  actual 
application  of  these  formula;,  the  intervals  between  the  obserA'ations 
not  being  very  large,  I^  Avill  be  positive  Avhen  /9' — /9„  and  sin  (©'—  K) 
haA'c  contrary  signs,  and  negatiA'C  Avhen  /9'  —  /9q  has  the  same  sign  us 
sin  (O'  —  K).  Hence,  Avhen  O'  —  A'  is  less  than  180°,  /•'  must  be 
less  than  R'  if  /9'  —  /9„  is  positive,  but  greater  than  R'  if  /9'  —  /9^,  is 
negative.  When  O'  —  .jfiT  exceeds  180°,  r' Avill  be  greater  than /i' 
if  /9'  —  /?(,  is  positive,  and  less  than  R'  if  /9'  —  /9q  is  negative.  We 
may,  therefore,  by  means  of  a  celestial  globe,  determine  by  inspection 
Avhether  the  distance  of  a  comet  from  the  sui*  is  greater  or  less  than 


DETER51INATION   OF   AN   OUBIT. 


243 


tliat  of  the  earth  from  the  sun.  Thus,  if  we  pass  a  great  circle 
tlirou(rh  the  two  extreme  observed  places  of  the  comet,  ;•'  must  bo 
greater  than  W  when  the  place  of  the  comet  for  the  middle  observa- 
tion is  on  the  same  side  of  this  great  circle  as  t'le  point  of  the 
ecliptic  which  corresponds  to  the  place  of  the  sun.  But  when  the 
middle  place  and  the  point  of  the  ecliptic  corresponding  to  the  place 
of  the  sun  are  on  opposite  sides  of  the  great  circle  passing  through 
the  first  and  third  places  of  the  comet,  r'  must  be  less  than  R'. 

85.  From  the  values  of  p'  and  /•'  derived  from  the  assumed  values 
t" 
P  =  —  and   Q  =  Tr'',  >vo  may  evidently  derive  more  approximate 

values  of  these  quantities,  and  thus,  by  a  repetition  of  the  calcula- 
tion, make  a  still  closer  approximation  to  the  true  value  of  p'.  To 
derive  other  expressions  for  P  and  Q  which  are  exact,  i)rovided  that 
/  and  p'  are  accurately  known,  let  us  denote  by  s"  the  ratio  of  the 
sector  of  the  orbit  included  by  r  and  r'  to  the  triangle  included  by 
the  same  radii-vectores  and  the  chord  joining  the  first  and  second 
places ;  by  s'  the  same  ratio  with  respect  to  r  and  r",  and  by  «  this 
ratio  with  respect  to  r'  and  >•".  These  ratios  s,  s',  s"  must  neces- 
sarily be  greater  than  1,  since  every  jjart  of  the  orbit  is  concave 
toward  the  sun.  According  to  the  equation  {30)„  we  have  for  the 
ureas  of  the  sectors,  neglecting  the  mass  of  the  ho'^y, 


2^"  V'P> 


i^'v% 


■z' 


I  p> 


and  therefore  we  obtain 


s"[rr'-]=T"^p, 


s'[rr"-]=T'^/p, 


s[,V']  =  T|/>.     (^46) 


Then,  since 


we  shall  have 
and,  consequently, 


[rV] 
[»t"] 


r     s 
n  =  -,  •  -, 

T      S 


n"  = 


t"    s' 


T        S 


P  = 


t"     s 


T       S- 


Substituting  for  8,  s',  and  s"  their  values  from  (46),  we  have 


(47) 


(48) 


o_o,,.[^v']  +  K]-K']  III' 


(49) 


244 


THEORETICAL   ASTRONOMY. 


The  auf^iilar  distance  between  tlic  perihelion  and  node  being  denoted 
by  0),  the  polar  equation  of  the  conic  section  gives 


P 


P 


1  -f  e  cos  (u  —  w), 


7  =  1  +  e  cos  (u'  —  Hi), 
^  =  1  +  e  cos  (it"  —  w). 


(50) 


If  we  multiply  the  first  of  these  equations  by  sin  [u'' —  w'),  the  second 
by  —  sin  [u"  —  u),  and  the  third  by  sin  («'  —  u),  add  the  products 
and  reduce,  we  get 


P 


P 


sin  (it"  —  u')  — ■'-,  sill  (m"  —  «)  -f  ^  sin  («'  —  u)  =  sin  {u"  —  it') 


P 


and,  since 


sin  («"  —  ?f)  +  sin  (it'  —  u)  ; 


sin  (u"  —  «')  =  2  sin  \  («"  —  it')  cos  J  (it"  —  it'), 

sin  (ii" —  i()  —  sin  ( u' —  it)  =  2  sin  A  (it" —  it')  cos  h  (it"  +  it' —  2u), 

the  second  member  reduce^:  to 

4  sin  ^  (it"  —  it')  sin  I  (it"  —  it)  sin  A  (it'  —  it). 
Therefore,  we  shall  have 

Arr'r'  sin  -^  (it"  —  i'')  sin  \  (it"  —  it)  sin  ^  (it'  —  i',) 


P 


r'r"  sin  (it" —  «')  —  rr"  sin  (it"—  it)  +  r/  sin  (it' —  it)* 


If  we  multiply  both  numerator  and  denominator  of  this  expression 

by 

2rr'r"  cos  ;\  (it"  —  it' )  cos  ^  (  ""  —  it)  cos  \  (it'  —  it), 

it  becomes,  introducing  [/v'],  [^■'■"],  and  [r'r"], 

_  Jr'/'Mn-'2,  M 1 

P  ~  [r'r"]  +  [r/]  —  [i-r"]  "  2rr'r"  cos  A  (it"— it')  cos  A  (it"— it)  cos  ^  (it'— it)' 

Substituting  this  value  of  jj  in  equation  (49),  it  reduces  to 


TT 


^  "~  W   ■  r,."  nns  I  r^i."  — 


ss"    rr"  cos  ^  (it"  —  it')  cos  \  (it"  —  it)  cos  \  (it'  —  it)' 


(51) 


86.  If  we  compare  the  equations  (47)  with  t  he  formula  (28)3,  we 
derive 


g"_1         .t'-t'"        ,(t»4-t'^')     rf/ 


•"      "^  *       yfcr'*      ■  rf< 


(52) 


DETERMIIiATION  OF   AN   ORBIT. 


245 


Consequently,  in  the  first  approximation,  we  may  take 


=  1. 


If  the  intervals  of  the  times  arc  not  very  unequal,  this  assumption 
will  differ  from  the  truth  only  in  terms  of  the  third  order  vith  respect 
to  the  time,  and  in  terms  of  the  fourth  order  if  the  intervals  are 
equal,  as  has  already  been  shown.  Hence,  we  adopt  for  the  first 
ai)proximation, 


q  =  rr", 


the  values  of  r  and  7"  being  computed  from  the  uncorrected  times 
of  observation,  which  may  be  denoted  by  i^,  fj,  and  tj'.  With  the 
values  of  P  and  Q  thus  found,  we  compute  »•',  and  from  this  p',  p, 
and  [)",  by  means  of  the  formula)  already  derived. 

The  heliocentric  places  for  the  first  and  third  obser'^ations  may 
now  be  found  from  the  formuke  (71)3  and  (72)3,  and  then  the  angle 
n"  —  u  between  the  radii- vectores  /•  and  r"  may  be  obtained  in 
various  ways,  precisely  as  the  distance  between  two  points  on  the 
celestial  sphere  is  obtained  from  the  spherical  co-ordinates  of  these 
points.     When  u"  —  it  has  been  found,  we  have 


iir 


sin  {xc' —  ?t')  =  -—  sin  («"  —  «), 


n"r" 


(53) 


sm  {u  —  u)  =  —J-  sm  \u  —  u) 


from  which  •%' 


W 


and  u'  —  u  may  be  computed.     From  these 


results  the  ratios  s  and  s"  may  be  computed,  an  ^  then  new  and  more 
aj)|)ro>-imate  values  of  P  and  Q.  The  value  oi'  u"  —  u,  found  by 
tiidng  the  sum  of  u"  —  u'  and  u'  —  ti  as  derived  from  (53),  should 
aifrce  with  that  used  in  the  second  members  of  these  equations, 
within  the  limits  of  the  erroi's  which  may  be  attributed  to  the 
logarithmic  tables. 

The  most  advantageous  method  of  obtaining  the  angles  between 
the  radii-vectores  is  to  find  the  position  of  the  plane  of  the  orbit 
directly  from  /,  I",  b,  and  b",  and  tlien  compute  u,  u',  and  a"  directly 
from  SI  and  i,  according  to  the  first  of  equations  (82),.  It  will  bo 
expedient  also  to  compute  r',  Z'  ai  i  b'  from  />',  //,  and  ,9',  and  the 
agreement  of  the  value  of  /■',  t»  ns  found,  with  that  already  obtained 
from  equation  (37),  will  check  the  accuracy  of  part  of  the  numerical 


246 


THEORETICAL   ASTRONOMY. 


calculation.  Further,  since  the  three  places  of  the  body  nuist  be  in 
a  plane  passing  through  the  centre  of  tlie  sun,  whether  F  and  Q  arc 
exact  or  only  approximate,  wa  nuist  also  have 

tan  h'  =  tan i  sin  (/'  —  Q,), 

and  the  value  of  i^  derived  from  this  equation  must  agree  with  that 
computed  directly  from  ()',  or  at  least  the  difference  should  not  cxcood 
what  may  be  due  to  the  unavoidable  errors  of  logarithmic  calcula- 
tion. 

We  may  now  compute  n  and  n"  directly  from  the  equations 


n  = 


r'r" 


sin  (ft" —  «') 
rr"  sin  {u" —  %i)^ 


„       rr  sni(?<'— lO  ,... 

rr   sm  {u  —  u) 


but  when  the  values  of  u,  d',  and  u"  are  those  which  result  from  the 
assumed  values  of  P  and  Q,  the  resulting  values  of  n  and  n"  \vill 
only  satisfy  the  condition  that  the  plane  of  the  orbit  passes  through 
the  centre  of  the  sun.  If  substituted  in  the  equations  (29),  they  will 
only  reproduce  the  assumed  values  of  P  and  Q,  from  which  they 
have  been  derived,  and  hence  they  cannot  be  used  to  correct  them. 
If,  therefore,  the  numerical  calculation  be  correct,  the  values  of  n 
and  n"  obtained  from  (54)  must  agree  with  those  derived  from  equa- 
tions (31),  within  the  limits  of  accuracy  admitted  by  the  logarithiuii' 
tables. 

The  differences  u"  —  u'  and  u'  —  u  will  usually  be  small,  and 
hence  a  small  error  in  either  of  these  quantities  may  considerably 
aft'ect  the  resulting  values  of  n  and  n".  In  order  to  determine 
whether  the  error  of  calculation  i,s  within  the  limits  to  be  expected 
from  the  logarithmic  tables  used,  if  we  take  the  logarithms  of  both 
members  of  the  equations  (54)  and  differentiate,  supposing  only  n, 


n' 


ti 


and  v!  to  vary,  we  get 


rfloge?^   =  —  cot(w"  —  li!)  dii' , 
d  lege  n"  =  +  cot  (?t'  —  u)  du'. 

Multiplying  these  by  0.434294,  the  modulus  of  the  common  system 
of  logarithms,  and  expressing  du'  in  seconds  of  arc,  we  find,  in  units 
of  the  seventh  decimal  place  of  common  logarithms, 

dlogn  =  —  21.055  cot  («"  —  v!)du', 
d  log  n"  =  +  21.055  cot  (V  —  u)  du'. 

If  we  substitute  in  these  the  differences  between  logn  and  log?i"  as 
found  from  the  equations  (64),  and  the  values  already  obtained  by 


DETERMINATION   OF   AN   ORBIT. 


247 


means  of  (31),  the  two  resulting  values  of  (hi'  should  agree,  and  the 
magnitude  of  du'  itself  will  show  whether  t)ic  error  of  calculation 
exceeds  the  unavoidable  errors  due  to  the  limited  extent  of  the 
logarithmic  tables.  When  the  agreement  of  the  two  results  for  n 
and  n"  is  in  accordance  with  these  conditions,  and  no  error  has  been 
made  in  computing  71  and  n"  from  P  and  Q  l)y  means  of  the  equa- 
tions (31),  the  accuracy  of  the  entire  cideulation,  both  of  the  (jiuiu- 
tities  which  depend  on  the  assumed  values  of  P  and  Q,  and  of  th<>^e 
which  are  obtained  independently  from  the  data  burnished  by  observa- 
tion, is  completely  proved. 

87.  Since  the  values  of  n  and  n"  derived  from  equations  (o4) 
cannot  be  used  to  cori'ect  the  assumed  values  of  P  and  Q,  from 
wliich  r,  r',  «,  u',  &c.  Jiave  been  computed,  it  is  evidently  necessary 
to  comj)ute  the  values  for  a  second  approximation  by  njcms  of  the 
series  given  by  the  equations  (26),,  or  by  means  of  the  ratios  «  and 
s".  The  expressions  for  n  and  n"  arranged  in  a  series  with  respect 
to  the  time  involve  the  differential  coefficients  of  r'  w  ith  respect  to  t, 
and,  since  these  are  necessarily  unknown,  and  cannot  be  coiiveniently 
determined,  it  is  plain  that  if  the  ratios  ■>>•  and  s"  can  be  readily  found 
from  r,  r' ,  v" ,  u,  u',  u" ,  and  r,  r' ,  z",  so  as  to  involve  the  relation 
between  the  times  of  observation  and  the  places  in  tiie  orbit,  the_,' 
may  be  used  to  obtain  new  values  of  P  and  Q  by  means  of  equations 
(48)  and  (51),  to  be  used  in  a  second  approximation. 

Let  us  now  resume  the  equation 

M=E-^-e^mE, 
or 

h{t—T) 


3 


and  also  for  the  third  place 

k{f—T) 


Subtracting,  we  get 
a* 


3 
a^ 


=  £  —  c  sin  E, 


^E"  —  esmE". 


E—2esm^  {E"  —  E)  cos  A  (E"  +  E). 


(55; 


This  equation  contains  three  unknown  quantities,  a,  e,  and  the  dif- 
ference E"  —  E.  We  can,  however,  by  means  of  expressions  in- 
volving r,  r",  u,  and  m",  eliminate  a  and  e.  Thus,  since  p-—a{l  — e^), 
we  have 


T'Vp  =-  a?V^l  —  e' {E"  —E—2esm^  {E"  —  E)  eos  J-  (£'■  -f-  E)).  (56) 


248  THEORETICAL   ASTIIONOMY. 

From  the  equations 

l/r  sin  }^v  =  VaCTfe)  sin  IE,  VV'  sin  {v"  --=  Va{\-\e)  sin  \E'\ 

Vr cos Iv  =  1/0(1— e)  cos ^E,  Vr"  cos ^i>"  =  Va{\  —  e)  cos  Ai,'", 

since  y"  —  v  =  u"  —  u,  we  easily  derive 

VW'  sin  i  (it"  —  tt)  =  aVV^'  sin  |  (i;"  —  E),  (57) 

and  also 

a  cos  I  (£"  —E)  —  ae  cos  ^  (-E"  +-£)=:  l/^  cos  ^  (tt"  —  it), 
or  

e  cos  I  (£"  +  £)  =  cos  I  (£"  -  £)  -  ^ '"''''''"'  ^'^'*'~''^.     (08) 

Substituting  this  value  of  e  cos^{E" -{-  E)  in  equation  (56),  we  get 

z'V'p  r=  aVl^^'  {E"  —  E—  sin  (E"  —  E)) 

+  2al/r^=~e^  sin  -i  (E"  ~  E)  cos  i  («"  —  u)  V^', 

and  substituting,  in  the  last  term  of  this,  for  aVl  —  e^,  its  value  from 
(57),  the  result  is 


t' I'p  =  aVl  —  e'  {E"  —E  —  sin  (E"  —  E))  +  rr"  sin  («"  —  u).   (59) 
From  (57)  we  obtain 

"^V     ^  ""i>         sin4(^"-£)      ' 

or 

/      rr"sin(it'^  — tt)      \''  1 

0=1   1  — e'  =  \2Viy'cosK^t"-«)^  i^sin-U (£"—£)' 
Therefore,  the  equation  (59)  becomes 

1    E"  — E—sin  {E"—E)j  [rr"] 


r'Vp 


'p  suiniE"—E) 


\  2l/rr"  cos  i  («"  —  «)/ 


(60) 


Let  x'  be  the  chord  of  the  orbit  between  the  first  and  third  places, 
and  we  shall  have 

x'^  :=  (r  +  r")^  —  4rr"  cos^ -K""  —  «). 

Now,  since  the  chord  x'  can  never  exceed  r  +  r",  we  may  put 

x'  =  (r  +  r")sinr',  (61) 

and  fi'om  this,  in  combination  with  the  preceding  equation,  we  derive 

2]/»V'  cos  ^  («"  —  u)  =  (r  +  r")  cos  /.  (62) 


DETERMINATION   OF  AN  ORBIT. 


249 


Substituting  this   value,   and   [7t"]  =  -7  k  _p,   in  equation   (60),   it 
reduces  to 


E"  —  E-mi{E"  —  E') 


f;;74  +  .7-=1.      (63) 


sin'-i(^"— -^')  (r  +  r")"  cosV'    ^'    '  a 

The  elements  a  and  e  are  thus  eliminated,  but  the  resulting  equation 
involves  still  the  unknown  quantities  W — A' and  ,s'.  It  is  neces- 
sary, therefore,  to  dci'ive  an  additional  equation  involving  the  same 
unknown  qn.antities  in  order  that  E"  —  E  maybe  eliminated,  and 
that  thus  the  ratio  h',  which  is  the  quantity  sought,  may  be  found. 
From  the  equations 

r=^a  —  ac  cos  E,  r"  =  a  —  ae  cos  E", 

we  get 

r"  +  r  =  2a  —  2ae  cos  A  {E"  +  E)  cos  A  {E"  —  E). 

Substituting  in  this  the  value  of  c  co^\{E" -\-  E)  from  (58),  we  have 

r"-\-r  =  2a  sin»  \{E"  —  E)  +  2V^'  cos  A  («"  —  u)  cos  J  (£"  —  E), 
and  substituting  for  sin^(JE^"—  E)  its  value  from  (57),  there  results 

But,  since 


^ 


2r" 


we  have 

r4-r"--. 
^  8'-'    (r-fr")'co8V 

from  which  we  derive 

8in'J(J5;"— ^)  = 


2prr"  cos'^  ^  (w"  —  tt) 
1 


2/^/ 1 \« 

■«'='  \2l/r^'cosK«"— «))' 


7  +  0-  +  »•")  cos  /  (1  -  2  sin'  \  {E"  -  E)\ 


1 


^'» 


sin''  y 


7?     (;r  +  r")'cosV         cosr' 


/ ' 


(64) 


which  is  the  additional  equation  required,  involving  E" —  -S/and  s' 
as  unknown  quantities. 

Let  us  now  put 

svn^\{E"—E) 


3  -pil 


E"—E  —  sm{E"  —  Ey 


m  =: 


,."^»noa9v" 


(r  +  )•")»  cosV 


(05) 


sin'  .J/ 


cos;* 


250 


TIIEOUKTICAL  ASTRONOMY. 


and  the  equations  (63)  and  (G4)  bcoomo 


m! 


'1,1 


1/  's"'^h'~'^' 


(66) 


When  tlie  vahie  of  y'  Is  known,  the  first  of  these  equations  will 
enable  us  to  determine  s',  and  henec  the  value  of  x*',  or  )i'n\^\{E" — E), 
from  the  last  e(juatiou. 

The  calculation  of  y'  may  be  facilitated  by  the  introduction  of  an 
additional  auxiliary  quantity.     Thus,  let 


tan/ 


^'-4 


(67) 


and  from  (62)  we  find 


2»/ 


rr" 


cos  /  =  cos  ;J  (n"  —  u)  \'-r~-jr  =  2  cos  },  (u"  —  ti)  cos'/  tan  ■/, 


or 

We  have,  also, 
which  gives 


cos  /  =  sin  2x'  cos  ^  («" —  «). 

x'"  =  (r  +  r"y  —  Arr"  cos^"  \  (u"  —  «), 
x'^  =  (r  —  r")"  +  4n-"  sin"  -}  («"  —  u). 


(68) 


Multiplying  this  equation  by  cos"  J  (ft" — u)  and  the  preceding  one 
by  sin"  |  {u"  —  v),  and  adding,  we  get 

x"  ^  (r  +  r")'  sin-  ^  (u"  —  ii)  +  (r  —  r"y  cos' '  («"  —  u).      (69) 

From  (67)  we  get 


cos''/ : 


sin^/: 


and,  therefore, 


r  +  r' 


,"» 


r  —  r" 
cos  2;?'  =  — j — jf, 

r  -\-  r 

so  that  equation  (69)  may  be  written 

.   ^  „.-  =  sin» /  =  sin' ^ («"  —  «)  +  cos' 2/'  cos' A (w"  —  u). 

We  may,  therefore,  put 

sin  /  cos  (t'  =  sin  ^  («"  —  t()> 

sin /  sin  G'  =  cos  |  («"  —  it)  cos  2/, 

cos  >»'  =  cos  -^  {u"  —  u)  sin  2/, 


(70) 


DETEIlMrNATIOK   OF   AN   OKHIT. 


251 


from  which  y'  may  he  derived  by  means  of  its  tiuigent,  .so  that  sin  y' 
shall  he  positive.  The  auxiliary  angle  G'  will  l)e  of  suhscquent  ii.so 
ill  (li'tcrmining  the  elements  of  the  orbit  from  the  linul  hypothesis  for 
Paml  q. 

88.  We  shall  now  consider  the  auxiliary  quantity  y'  introduced 
into  the  first  of  ecpiatious  (G(i).     For  brevity,  let  us  put 


and  we  shall  have 


This  gives,  by  differentiation, 


g^\{E"-E), 


, sin^.7 

^  ~  2(/  —  sin^' 


^y       o     i     J  4  sin' 7^7 

-T-  —  3  cot  ff  dg  —  ,y •f~f-, 

y  ^U  —  SHI  2g 


or 


dl 


By'  cot  g  —  4^  cosec  g. 


The  last  of  equations  (65)  giv^es  x'  =  a'nv l(/,  and  hence 

^^2  cosec j7. 
Therefore  wc  have 

dy'      6y'  cos  g  —  Sy"  __  3  (1  —  2x'')  y'  —  4?/'» 


dx' 


sur  g 


2a;' (!—«') 


It  is  evident  that  we  may  expand  y'  into  a  series  arranged  iu  refer- 
ence to  the  ascending  powers  of  x',  so  that  we  shall  have 

y'  =  a  +  ,3.r'  4-  yx"  +  oV^  +  sx"-  +  Zx'^  +  &c. 

Differentiating,  we  get 


dxf 


dx' 


^  =  /9  +  2yx'  +  Sdx"  +  i^x"  +  5:x'*  +  &c., 

dy' 


and  substituting  for  j-j  the  value  already  obtained,  there  results 

2iSx  +  (4r  —  2/3)  x"  +  (6'5  —  4y)  x"  +  (8^  —  6<J)  a;'*  +  (IOC  —  8s;  .r'«+  &c. 
=  (3tt  —  4a*)  +  (3;5  —  6a  —  8a,?)  x'  +  (3^  —  6,5  —  i,J'  —  80^)  •i'" 
+  (35  —  6r  —  8/3,-  —  8a<5)  .r''  +  (3.'  —  6'J  —  4f  —  8,3«5  —  8a£)  x'* 
+  (3:  —  6s  —  8r<5  —  8/3£  —  8a:)  x"  +  &c. 

Since  the  coefficients  of  like  powers  of  z'  must  be  equal,  we  have 

3a  —  4a«  =  0,  3/3  —  6a  —  8a/3  =  2/3, 

3r  —  6/3  —  4/3»  —  Sa;- =  2  (2r  — /3),  &c. ; 


252 

ami  hence  wc  derive 


THEORETICAL   ASTUONOMY. 


a- I. 


'  —  33«ri75» 

Therefore  we  have 


'•  —  3l6!)0575> 


r?5» 


h  —    2" 


„  —      I  9  I  3  U  0  -2  1 


I         1  !t  I  ;t  u  (I  v!  I      /6     I     A'p  (  7 1  ^ 


If  Ave  multiply  through  by  y,  and  put 


we  obtain 


ft'___'(    ya     I         f)'.'      ,.'3     1        13  84    ^4     I         ft  0  08  8     ^4 
•»   —  3  5*     T^   |51;.  •'^     T^  0737S*     T^  ial 'J3  IB* 

I      ^  3  8  J  7  8  (I  I  8      ,,.'6  _l      ,«rp 


.^Oy_  5   _!_.,' __,.'. 


(72) 
(78) 


Combining  this  with  the  seeond  of  equations  (66),  the  result  is 


m' 


If  we  put 
wc  shall  have 


'fl^y'+^-^—i+Z  +  f'. 


,'  = 


m 


1+/  + 


t"" 


(74) 


i^o  y— «'^_,' 


But  from  the  first  of  equations  (66)  we  get 


m 


and  therefore  we  have 


y 


-=,'»(/-!); 


»'!  r»' 


(76) 


As  soon  as  ^'  is  known,  this  equation  will  give  the  corresponding 
value  of  s' . 

Since  f '  is  a  quantity  of  the  fourth  order  in  reference  to  the  diftbr- 
ence  \  {E"  —  E),  we  may  evidently,  tor  a  first  approximation  to  the 
value  of  yj',  take 


m 


and  with  this  find  s'  from  (75),  and  the  corresponding  value  of  x' 
from  the  last  of  equations  (66).  With  this  value  of  x'  we  find  the 
corresponding  value  of  f ',  and  recompute  tj',  s',  and  x' ;  and,  if  the 


DKTKRMINATION    OF    AN    OUniT. 


253 


valiit'  of  c'  (Icrivi'd  from  the  la(?t  value  of  x'  ditlers  from  that  already 
used,  the  operation  must  ho  ropcatnL 

It  will  he  ohservcd  that  the  iSeries  (72)  for  ?'  converges  with  j^reat 
rapidity,  and  that  for  E"  —  E^^dA°  the  term  containing  .r'"  amounts 
to  only  one  unit  of  the  seventh  decimal  place  in  the  value  of  ^'.  Tahle 
XIV.  gives  the  values  of  c'  corresponding  to  values  of  .i-'  from  0.0 
to  O.-.i,  or  from  E"~E=0  to  E"  —  E  V.V1°  GO'.G.  Should  u 
wise  occur  in  which  E"  —  E  exceeds  this  limit,  the  expression 

,_ 8in-^  \  jE"  -  E) 

y  —  E"—E  —  mWE"—E) 

may  then  be  computed  accurately  hy  means  of  the  logarithmic  tables 
ordinarily  in  use.  An  approximate  value  of  x'  may  be  easily  ibund 
with  which  y'  may  be  com])uted  from  this  equation,  and  then  ^'  from 
(7''5).  With  the  value  of  ? '  thus  found,  r/  may  be  comi)uted  from 
(74),  and  thus  a  more  approximate  value  of  x'  is  immediately 
obtained. 

The  ecpiation  (7o)  is  of  the  third  degree,  and  has,  therefore,  three 
loots.  Since  .s'  is  always  positive,  and  cannot  be  less  than  1,  it 
Mlv.ws  from  this  equation  that  tj'  is  always  a  positive  (quantity.  The 
equation  may  be  written  thus : 


r;V 


h' 


0, 


ami  there  being  only  one  variation  of  sign,  there  can  be  only  one 
positive  root,  which  is  the  one  to  be  adopted,  the  negative  roots  being 
excluded  by  the  nature  of  the  problem.  Table  XIII.  gives  the 
values  of  log.s'^  corresponding  to  values  of  tj'  from  3y'=0  to  jy'=^0.6. 
Wliou  r/  exceeds  the  value  O.G,  the  value  of  s'  must  be  found  directly 
from  the  equation  (75). 

89.  We  are  now  enabled  to  determine  whether  the  orbit  is  an 
ellij)sc,  parabola,  or  hyperbola.  In  the  ellipse  x  =  sin"'^  \  {E"  —  E) 
is  positive.  In  the  parabola  the  eccentric  anomaly  is  zero,  and  hence 
a'--=0.  In  the  hyperbola  the  angle  which  we  call  the  eccentric 
anomaly,  in  the  case  of  elliptic  motion,  becomes  imaginary,  and 
lionce,  since  fi\n\{E"  —  ^)  will  be  imaginary,  ;«'  nuist  be  ncgativ^e. 
It  follows,  therefore,  that  if  the  value  of  x'  derived  from  the  equa- 
tion 

,      m'       ., 


is  positive,  the  orbit  is  an  ellipse ;  if  equal  to  zero,  the  orbit  is  a 
parabola ;  and  if  negative,  it  is  a  hyperbola. 


254 


theore:tical  astronomy. 


For  the  case  of  ])arabo]ic  motion  wc  liave  x'  =  0,  and  the  second 
of  equations  (6(3)  gives 

„      m' 
^'^If-  176) 


If  we  eliminate  s'  by  raoaiu*  of  both  equations,  since,  in  this  case, 
y'  =  i  we  get 

Substituting  in  this  tl' .'  vahies  of  m,  and  I  given  by  (65),  we  obtain 


'»_' 


wliich  gives 


(r  +  r")i 


Qt' 


(^  +  r")^ 


3  sin  1/  cos  •/  -f  4  sin"  A/, 


6  sin  hy'  CDs'*  Ay'  +  2  sin'  A/, 


or 


OM-r") 
This  may  evidently  be  written 


— ;  =-^  (sin  1/  +  cos  Jr')*  +  (sin  \/  —  cos  A/)' 


^-' 


6r  , 

rTT^Ti  ='-"  '^^  +  siii/)i  :f  (1  —  smr')^' 
{r-\-r  )a 

the  upper  sign  being  used  when  y'  is  less  than  90°,  and  the  lower 
sign  when  it  exceeds  90°.  Multiplying  through  by  (."  +  r")^-,  and 
replacing  {r  -'-  /•")  sin  y  by  '/.,  we  ol)tain 

6r'  n^  (r  +  r"  +  x)^  qr  (r  +  r"  -  x)!, 

which  is  identical  with  the  equation  (06)3  for  the  special  case  of 
parabolic  motion. 

Since  x'  is  negative  in  the  case  of  hyperbolic  motion,  the  vai  u;  nf 
?'  determined  by  the  series  (72)  will  hv  diflerent  from  that  in  the 
case  of  elliptic  motion.  Table  XIV.  gives  the-  value  of  ?'  corre- 
sponding to  both  forms;  but  when  x'  exceeds  the  limits  of  this  tablo, 
it  ^vill  be  necessary,  in  the  case  of  the  hyjx'rbola  also,  to  comj)Utc  llie 
value  of  ?'  directly,  using  additional  terms  of  the  series,  or  wc  may 
modify  the  expression  for  y'  in  terms  of  ii"'  and  £!  so  as  to  lie 
apjtlicablo. 

If  we  compare  equations  (44),  and  (56)i,  we  get 

tan  hE---  V~-l  tan  hF; 


DETERMINATIOX   OF   AN   ORBIT.  255 

and  lionce,  from  (58),, 

a  -j-  1 

Wc  have,  also,  by  comparing  (Go),  with  (41)i,  since  a  is  negative  in 

the  liypcrbola, 

ff'  + 1 


cos-E 


which  gives 
Now,  since 


2ff   ' 

■'      1    /- 


sin  E  =  — 7c —  V  —  1. 


ill  which  c  is  the  base  of  Naperian  logarithms,  we  have 

E  l/— ~i  =:  log„  (cos  E  +  l/^'  sin  E), 
which  reduces  to 


or 


£=  r —  1  loofe"". 


By  means  of  these  relations  between  E  and  <t,  the  expression  for  ?/' 
may  i)o  transformed  so  as  not  to  involve  imaginary  quantities.  Thus 
wo  have 

E"  -  E  r=:  (logeV  ~  loge  t)  V'^^l  ==-"  V^^l  log,  ~, 

sin  (^"  —  E)  =  sin  i:"  cos  ^  —  cos  E"  sin  ^  -=  1^^^^V^=1. 


Fioni  the  value  of  cosi?  we  easily  derive 
ff-1 


luul  hence 


siu^E— - — ~V—1, 

2i/<r 


cos  .',itJ  = 


ff+_l 


(T     o 


sin  A  (-B"  ~  i;)  -=  — ----.:  v'ZTT. 


Tliorefore  the  expression  for  y'  becomes 

U'  -  ay 


Cl/ffff")'  log.''--  -  4  V'ffa''  (-7'"—  ff') 


256 


THEORETICAT,  ASTRONOMY. 


Since  the  auxiliary  quantity  a  in  the  hyperbola  is  always  positive, 
let  urf  now  put 


and  we  have 


-^', 


\iA-^ 


from  which  y'  may  be  derived  when  A  is  known. 
^\^c  have,  further, 

sin=  \  (E"  —  E)  =  i{l-  cos -\  (E" -  ES)  =  \i\ 

and  tlierefore 


(77) 


2  l/ff<7"  ) ' 


4  Va,s" 


or 


a;'  = 


^(-^^-^r- 


(78) 
(79) 


These  expressions  for  y'  and  a;'  enable  us  to  find  ^'  when  x'  exceeds 
the  limit';  of  the  table.  Thus,  we  obtain  an  approximate  value  of  x' 
by  putting 


m 


V  = 


+  f 


from  whieli  we  first  find  41'  and  then  x'  from  the  second  of  equations 
(66).     Then  we  compute  A  from  the  formula  (79),  which  gives 


.4  =  1  —  2*'  +  2Vx''  —  x', 


(80) 


y'  from  (77),  and  ?'  from  (73).  A  repetition  of  the  calculation,  using 
the  value  of  c'  lluis  found,  will  give  a  still  closer  approximation  to 
the  correct  values  of  x'  and  s' ;  and  this  process  should  bo  continued 
until  ?'  remains  unchanged. 

90.  The  formulte  for  the  calculation  of  s'  will  evidently  give  the 
value  of  H  if  we  use  7,  ?•',  /•",  u',  and  «",  the  necessary  changes  in  the 
notation  being  indicated  at  once;  and  in  the  same  manner  using  r", 
r,  r',  u,  and  «/,  we  obtain  s".  From  the  values  of  a  and  .s"  thus 
found,  more  accurate  values  of  P  and  Q  may  be  computed  by  means 
of  the  equations  (48)  and  (51).  We  may  remark,  however,  that  it' 
the  times  of  the  observations  have  not  been  already  corrected  for  the 


DETERMINATION   OF   AN   ORBIT. 


257 


tinie  of  aberration,  as  in  the  ease  of  the  determination  of  an  unknown 
orbit,  this  correction  may  now  he  applied  as  determined  by  means  of 
the  values  of  p,  p',  and  p"  already  obtained.  Thus,  if  t^,  t/,  and  f^" 
are  the  uncorrected  times  of  observation,  the  corrected  values  will  be 


f: 


Cp  sec  ,3, 
Cy  sec  /5', 
(.y'sec/3", 


(81) 


in  which  log  C  — 7.760523,  expressed  in  parts  of  a  day;  and  from 
these  values  of  t,  t',  t"  we  recompute  r,  r',  and  z",  which  values  will 
require  no  further  correction,  since  p,  p' ,  and  p" ,  derived  from  the 
first  approximation,  are  sufficient  for  this  purpose.  With  the  new 
values  of  P  and  Q  we  recompute  r,  r',  r",  and  u,  u',  u"  as  before, 
and  thence  again  P  and  Q,  and  if  the  last  values  differ  from  the  pre- 
ceding, we  proceed  in  the  same  manner  to  a  third  approximation, 
which  will  usually  be  sufficient  unless  the  interval  of  time  between 
the  extreme  observations  is  consideiable.  If  it  be  found  necessarv 
to  proceed  further  with  the  approximations  to  P  and  Q  after  the 
calculation  of  these  quantities  in  the  third  approximation  has  been 
ofFeoted,  instead  of  employing  these  directly  for  the  next  trial,  we 
may  derive  mox'c  accurate  values  from  those  already  obtained.  Thus, 
let  X  and  _?/  be  the  true  values  of  P  and  Q  respectively,  with  which, 
if  the  calculation  be  repeated,  we  should  derive  the  same  values  again. 
Let  A,r  and  ai/  be  the  differences  betsveen  any  assumed  values  of  x 
and  y  and  the  true  values,  or 


:  X  +   ^^) 


yo^y  +  ^y- 


Then,  if  we  denote  by  a;„',  i/q'  the  values  which  result  by  direct  cal- 
culation from  the  assumed  values  Xq  and  y^,  we  shall  have 

Expanding  this  function,  we  get 

»o'  -  ^0  =f(^^,  y)  +  ^^^  +  B^y  +  C^x'  +  Z>A.r  ^y  -f-  Ei.f  +...., 

and  if  a.?'  and  a^  are  very  small,  we  may  neglect  terms  of  the  second 
order.  Further,  since  the  employment  of  x  and  y  will  reproduce  the 
same  values,  we  have 

/(^,y)  =  o, 

f'ud  hence,  since  a.x'  ==  a;„  —  x  and  a?/  ==  ?/« —  Vi 


x^^A  (xo  —  x)-\-B{]i^~  y). 

17 


258 


THEORETICAL   ASTRONOMY. 


In  a  similar  manner,  we  obtain 

2/0'  —  2/0  =  ^'  (•*'o  —  ^)  +  -S'  (yo  —  2/). 

Let  us  now  denote  the  values  resulting  from  the  first  assumption  for 
P  and  Q  by  P,  and  (^j,  those  resulting  from  P„  (^^  by  Pj,  ^^o,  and 
from  Pj,  ^2  ^y  A>  ^3J  ^"<^)  further,  let 


P,-P,=a', 


Then,  by  means  of  the  equations  for  xj  —  Xq  and  y^'  —  7/0,  we  shall 
have 

a  ^A(P-x)  +  B(Q-y), 


a 


:A(P,-x)  +  B{Q,- 
AiP,-x)  +  B(Q^- 


h  =A'(P-x)  +  B'(Q-y), 
■y),  b'^--^'^A'{P-x)  +  B\Q,~y), 

■y),  h"^A'{P-x)  +  B(Q-y). 

If  we  eliminate  A,  B,  A',  and  B'  from  these  equations,  tLj  results 
are 

P(a'b"  —  a"b')  +  P,  (a"b  —  ah")  +  P,  (aV  —  a'b) 


x  = 


y  = 

from  which  we  get 


(„'6"  —  a"b')  +  ia"b  —  ab")  +  (ab'  —  a'b)       ' 

Qia'b"-a"b')  +  Q,  (a"b  —  ab")  +  ^,  («&'  -  a'6) 
( «'6"  —  a"6')  +  (cd'u  —  ab")  +  (a6'  —  a'b)      ' 


.r  =  P 


fa"  +  a')  (n'b"  —  a"b')  +  a"  (a"b  —  ah' 


) 


y=Q: 


"      {a'b"—  a"b')  -f  (a"6  —  a6")  +  {ab'  —  a'b)' 
(b"  +  b')  (a'b"  —  a"b')  4-  b"  (a"b  -  a6") 


(82) 


{a'b"—  a"b')  +  (a"6  —  ab")  +  (a6'—  a'6)" 


In  the  numerical  application  of  these  formukc  it  will  be  more  con- 
venient to  use,  instead  of  the  numbers  /■•,  P,,  P^,  Q,  §„  etc.,  the  loga- 
ritlnns  of  these  cjuantities,  so  that  a  =  log  Pj —  log  P,  b  --  log  C^,— -  log  Q, 
and  similarly  for  a',  b',  a",  b", — which  may  also  be  expressed  in 
units  of  the  last  decimal  place  of  the  logarithms  employed, — and  we 
shall  thus  obtain  the  values  of  log  a:  and  log?y.  With  these  values 
of  log.c  and  log^  for  log  P  and  log  Q  respectively,  we  proceed  with 
the  final  calculation  of  r,  r',  r",  and  m,  ?t',  u". 

When  the  eccentricity  ic  small  and  the  intervals  of  time  between 
the  observations  are  not  very  great,  it  will  not  be  necessary  to  employ 
the  equations  (82);  but  if  the  eccentricity  is  considerable,  and  if,  in 
addition  to  this,  the  intervals  are  large,  they  will  be  required.  It 
may  also  occur  that  the  values  of  P  and  Q  derived  from  the  Ifwt 
hypothesis  as  corrected  by  means  of  these  formula:,  will  differ  so 


DETERMIXATIOX   OF    AN   ORBIT. 


259 


much  from  the  vahies  fouiul  for  x  and  ij,  on  account  of  the  neglected 
terms  of  the  second  order,  that  it  will  be  necessary  to  recompute  these 
(jiiantities,  using  these  last  values  of  F  and  Q  in  connection  with  the 
three  preceding  ones  in  the  numerical  solution  of  the  equations  (82). 

91.  It  remains  now  to  complete  the  determinstion  of  the  elements 
of  the  orbit  from  these  final  values  of  P  and  Q,  As  soon  as  Q,,  i, 
and  u,  u',  u"  have  been  found,  the  remaining  elements  may  lie  de- 
rived by  means  of/",  r',  and  u' — u,  and  alsu  from  >•',  ?•",  and  u" — u' ; 
or,  which  is  better,  we  will  obtain  them  from  the  extreme  places,  and, 
if  the  approximation  to  P  and  Q  is  com})lete,  the  results  thus  found 
will  agree  with  those  resulting  from  the  combination  of  the  middle 
place  with  either  extreme. 

We  must,  therefore,  determine  s'  and  x'  from  r,  r",  and  u"  —  u, 
by  moans  of  the  formulte  already  derived,  and  then,  from  the  second 
of  equations  (46),  we  have 

y  =  ('''-'-"''"f"— >)',  (83) 

from  which  to  obtain  p.    If  we  compute  .s  and  s"  also,  we  shall  have 

/  sr'r"  sin  («"  —  «')  \ ^      /  /'rr' sin  (u'  —  u)\^ 
P=\- r )=^\ V )' 

and  the  mean  of  the  two  values  of  ^j  obtained  from  this  expression 
should  agree  with  that  found  from  (83),  thus  checking  tlio  calcula- 
tion and  showing  the  degree  of  accuracy  to  w'hicli  the  approximation 
to  P  and  Q  has  been  carried. 
The  last  of  equations  (6o)  gives 

sh\\(E"—E)=\/x',  (84) 

from  which  E"  —  E  may  be  computed.  Then,  from  equation  (57), 
since  c  =  sin  if,  we  have 


COS^P 


rr 


(85) 


■  sin -](-£;"— ^') 
for  the  crdculatiou  of  a  cos  f.     But  }>  =  «(1  —  r)  =  a  cos"  <f,  whence 


P 

cos  0  =  —--- — , 

a  cos  ^ 


(86) 


which  may  be  used  to  determine  <p  when  c  is  very  nearly  equal  to 
unity;  and  then  e  may  be  found  from 

e  =  l  — 2  8inH45°~^Y'). 


260 


THEORETICAL  ASTRONOMY. 


The  equations  (50)  give 


P 


e  cos  (u  —  w)  =  _  —  1, 
r 

e  cos  (u"  —  (1})=:^  —  1, 

and  from  these,  by  addition  and  sul  'Taction,  we  derive 

2e  cos  A  iu"  —  u)  cos(A  (u"  -f-  «)  —  w)  =■??  -\-^  —  2, 


2e  sin  i  (v"  —  u)  sin  (i  («"  -j-  «)  —  <")  ^—  -  —  zrT> 


r      r 

r      ¥" 


(87) 


by  means  of  which  e  and  u)  may  be  found. 
Since 


cos  2x' 


2Vri 


j> 


we  have 


•  +  r"' 


si°2/---qr-,7r' 


2;; 


^  +  ^-2=:- 

'■       '"^  Vrr"ii\n2-/: 

J)       j>  2j^cot2;^' 


--% 


r      r 
and  from  equations  (70), 

,      sin  ^  (ii"  —  u)  tan  G' 


cot  2/ 


cosy 


sin  2/ : 


cosy- 


cos  i  (tt" —  v) 


Therefore  the  formulae  (87)  reduce  to 


e  sin  (w  —  ^  iu"  +  ^())  = rT=r,  t^"  ^'' 


e  cos  (w  —  ^  (it"  +  ''0) 


P_ 

cos  /  V  rr' 
P 


(88) 


sec  i  (u"  —  w;, 


cos  /''  Vrr" 

from  which  also  e  and  <«  may  be  derived.     Then 

sin  <p  =  e, 

and  the  agreement  of  cos  f  as  derived  from  this  value  of  <p  Avith  that 
gi\'en  by  (86)  will  serve  as  a  further  proof  of  the  calculation.  The 
longitude  of  the  perihelion  will  be  given  by 

or,  when  i  exceeds  90°,  and  the  distinction  of  retrograde  motion  is 
adopted,  by  ;r  =  J^  —  o). 


DETERMINATION  OF   AN  OBBIT. 

To  find  a,  we  have 


a  = 


p         (a  cos  <p)'* 


cos'  ^  p 

or  it  may  be  computed  directly  from  the  equation 

^2 


a  = 


4s" rr"  cos'^  A  («"  —  it)  sinM  (£"  —  E)' 


261 


(89) 


which  results  from  the  substitution,  in  the  last  term  of  the  preceding 
equation,  of  the  expressions  for  a  cos  (f  and  j)  given  by  (83)  and  (85). 
Then  for  the  mean  daily  motion  we  have 


We  have  now  only  to  find  the  mean  anomaly  corresponding  to  any 
epoch,  and  the  elements  are  completely  determined.  For  the  true 
anomalies  we  have 


v=^u 


■■  u 


y"  ^  u"  —  u> ; 


and  if  we  compute  r,  r',  r"  from  these  by  means  of  the  polar  equa- 
tion of  the  conic  section,  the  results  should  agree  with  the  values  of 
the  same  quantities  previously  obtained.     Accoi'ding  to  the  equation 

(45)„  we  have 

tan  IE  =  tan  (45°  —  \<p)  tan  Av, 
.    tanljS'  =  tan  (45°  —  l<p)  tan  >',  (90) 

tan  \e"  =  tan  (45°  —  \<p)  tan  y', 

from  which  to  find  E,  E',  and  E".  The  difference  E"  —  E  should 
agree  with  that  derived  from  equation  (84)  within  the  limits  of 
accuracy  afforded  by  the  logarithmic  tables.     Then,  to  find  the  mean 

anomalies,  we  have 

M  =E  —  esmE, 

M'  =E'  —  esmE',  (91) 

M"^E"— e  sin  E"; 

and,  if  il/g  denotes  the  mean  anomaly  corresponding  to  any  epoch  T, 

we  have,  also, 

M,  =  M  ~ii{t  —T) 
=  M'~iJ.{t'--T) 
=  M"—ix{t'  —  T), 

in  the  application  of  which  the  values  of  t,  t',  and  t"  must  be  those 
which  have  been  corrected  for  the  time  of  aberration.     The  agree- 


I 


262 


THEORETICAL   ASTRONOMY. 


mcnt  of  the  tlu'ce  values  of  M^  will  be  a  final  test  of  the  accuracy  of 
the  entire  calculation.  If  the  final  values  of  P  and  Q  are  exact, 
this  proof  will  be  comj)letc  within  the  limits  of  accuracy  adniittod 
by  the  logarithmic  tables. 

When  the  eccentricity  is  such  that  the  equations  (91)  cannot  be 
solved  with  the  requisite  degree  of  accuracy,  we  nuist  proceed  accord- 
ing to  the  methods  already  given  for  finding  the  time  from  the  i)eri- 
helion  in  the  case  of  orbits  differing  but  little  from  the  parabola. 
For  this  purpose,  Tables  IX.  and  X.  will  be  employed.  As  soon  as 
V,  v',  and  v"  have  been  determined,  we  may  find  the  auxiliary  angle 
V  for  each  observation  by  means  of  Table  IX.;  and,  with  Fas  the 
argument,  the  quantities  3f,  31',  31"  (which  are  not  the  mean  anoma- 
lies) must  be  obtained  from  Table  VI.  Then,  the  perihelion  distance 
having  been  computed  from 

P 

we  shall  have 


T=t- 


a  ^1 


-\-e 


•=:<' 


3I'q^l'    2 


-t-e 


■  t"— 


Ell. 


l  +  e' 


(92) 


in  which  log  CJ,  =  9.96012771  for  the  determination  of  the  time  of 
perihelion  passage.  The  times  t,  t',  t"  must  be  those  which  have 
been  corrected  for  the  time  of  aberration,  and  the  agreement  of  the 
three  values  of  T  is  a  final  proof  of  the  nunu  ical  calculation. 

If  Table  X.  is  used,  as  soon  as  the  true  anomalies  have  been  found, 
the  corresponding  values  of  log^  and  log  C  must  be  derived  from 
the  table.     Then  w  is  computed  from 


tan 


,     tan  ^v    \  1  -f-  9t; 


5(1 +e)' 


and  similarly  for  i«'  and  xc"  ;  and,  with  these  as  arguments,  we  derive 
31,  31',  31"  from  Table  VI.     Finally,  we  have 


T=t- 


MBq^ 


'■D'A 


cy^'.a^de) 


t'- 


M'B'q^ 


cy^-^a  +  ^e) 


t"- 


M"B"q^ 


(93) 


for  the  time  of  perihelion  passage,  the  value  of  CJ,  being  the  same  as 
in  (92). 

When  the  orbit  is  a  parabola,  c  ^=  1  and  p  =  2q,  and  the  elements 
q  and  <o  can  be  derived  from  r,  r",  u,  and  u"  by  means  of  the  equa- 


DETERMINATION   OF   AN  ORBIT. 


263 


tions  (76),  (83),  and  (88),  or  by  means  of  the  formuhc  already  given 
for  the  special  case  of  i)arabolic  motion. 

92.  Since  certain  quantities  which  arc  real  in  the  ellipse  and  para- 
bola become  imaginary  in  the  case  of  the  hyi)erbola,  the  formula; 


,if 


already  given  for  determining  the  elements  froni  r,  r",  i<,  and 
require  some  modification  Avhen  applied  to  a  hyperbolic  orbit. 

When  s'  and  x'  have  been  found,  /),  c,  and  lo  may  be  derived  from 
equations  (83)  and  (87)  or  (88)  precisely  as  in  the  case  of  an  elliptic 
orbit.     Since  x'  =  sin"  \  {E"  —  E),  we  easily  find 


sin  I  (E"  —  E)^2  V  x'  —  x'\ 
and  equation  (85)  becomes 
a  cos  (p 


sin  I  (u" —  \i)  Vrr" 
2l/.7=¥^ 


(94) 


But  in  the  hyperbola  x'  is  negative,  and  hence  V  x'  —  x''^  will  be 
imaginary ;  and,  further,  comparing  the  values  of  p  in  the  ellipse 
and  hyperbola,  we  have  cos'"^  =^  —  tan^;!-,  or 


cos  ^  =  V  —  1  tan  4. 
Therefore  the  equation  for  a  cos  (p  becomes 

m\^{u"—u)\/r/' 


a  tan  4 


2  V  x"' 


(95) 


if  a  is  considered  as  being   positive,  from  which  a  tan  1^  may  be 
obtained.     Then,  since  p  =  a  tan'  4^,  we  have 


tan  4 


P 


a  tan  4. 


(96) 


for  the  determination  of  t^,  and  the  value  of  e  computed  from 

e  =  sec  i^  =:  V^l  +  tan'^4 

should  agree  with  that  derived  from  equation  (88).  When  e  differs 
Init  little  from  unity,  it  is  conveniently  and  accurately  computed 
from 

e  =  1  +  2  sin^  ^^  sec  4.. 


The  value  of  a  may  be  found  from 


tj=jjcot'4  = 


(a  tan  4)^ 
P 


(97> 


2fi4 


TIIEORETICAI.   ASTRONOMY. 


or  from 


a  =  zi 


mi"  rr"  cos"  \  («"  ~  u)  (x"  —  z')' 


which  is  derived  directly  from  (89),  observing  that  the  elliptic  senii- 
truii.sverse  axis  becomes  negative  in  the  case  of  the  hyperbola. 

As  soon  as  to  has  been  found,  wc  derive  from  it,  «',  and  u"  the 
corresponding  values  of  v,  v',  and  v",  and  then  comi)ute  the  values 
of  F,  F',  and  F"  by  means  of  the  formula  (57), ;  after  which,  by 
means  of  the  equation  (09),,  the  corresi)onding  values  of  A',  N' ,  and 
N"  will  be  obtained.  Finally,  the  time  of  perihelion  passage  will 
be  given  by 


T=^i 


3 


3  3 

N^t'-~N'  =  f-~N" 


wherein  log;,„Z;  --  7.8733G575. 

The  cases  of  hyperbolic  orbits  are  rare,  and  in  most  ol  iliose  which 
do  occur  the  eccentricity  will  not  differ  much  from  that  of  the  para- 
bola, so  that  the  most  accurate  determination  of  T  will  be  effected  by 
means  of  Tables  IX.  and  X.  as  already  illustrated. 

93.  ExAMPiiE. — To  illustrate  the  application  of  the  principal  for- 
mula; which  have  been  derived  in  this  chapter,  let  us  take  the  follow- 
ing observations  of  Earynomc  @  : 


Ann  Arbor  M.  T.  (w'a 

1863  Sept.  14  15*  53"*  37'.2  1»    0"  44'.91 

21     9  46    18.0  0  57      3.57 

28    8  49    29.2  0  52    18.90 


+  9°  53'  30".8, 
9    13    5  .5, 

+  8    22     8  .7. 


The  apparent  obliquity  of  the  ecliptic  for  these  dates  was,  respect- 
ively, 23^^  27'  20".75,  23°  27'  20".71,  and  23°  27'  20".65 ;  and,  by 
means  of  those,  converting  the  observed  right  ascensions  and  declina- 
tions into  apparent  longitudes  and  latitudes,  we  get — 


Ann  Arbor  M.  T. 

Longitude. 

Latitude. 

1863  Sept.  14  15*  53'" 

37'.2 

170  47;  2,1"  m 

+  3°    8'43".19, 

21     9  46 

18.0 

16    41  36  .20 

2    52  27  .4(3, 

28    8  49 

29.2 

15    16  56  .35 

+  2    32  42  .98. 

For  the  same  dates  we  obtain  from  the  American  Nautical  Almanac 
the  following  places  of  the  sun : 


NUMERICAL   EXAMPLE. 


True  Longitude, 

Latitude. 

loRi^o- 

172°    ^'42".l 

—  0.07 

0.0022140, 

178    37  17  .2 

+  0.77 

0.001  :wr)7, 

185    2f5  r,4  .8 

+  0.H7 

0.0005174. 

Since  the  elements  are  supposed  to  be  wholly  uiikuown,  tlie  pliioes 
of  tlie  planet  nuist  l)c  correeted  for  the  aberration  of  the  fixed  stars 
as  given  by  ecpiations  (1).  Thus  we  find  for  the  correetions  to  be 
applied  to  the  longitudes,  respeetively, 


- 18".48, 
and  for  the  latitudes, 
+  0".47, 


—  19".49, 


+  0".30, 


20".8, 


+  0".14. 


When  these  corrections  arc  applied,  we  obtain  the  true  places  of  the 
planet  for  the  instants  when  the  light  was  emitted,  but  as  seen  from 
the  places  of  the  earth  at  the  instants  of  observation. 

Next,  each  place  of  the  sun  must  be  reduced  from  the  centre  of 
tlio  earth  to  the  point  in  which  a  line  drawn  from  the  planet  through 
the  place  of  the  observer  cuts  the  plane  of  the  ecliptic.  For  this 
purpose  we  liave,  for  Ann  Arbor, 


/  =  42°  5'.4, 


log /)„  =  9.99935; 


and  the  mean  time  of  observation  being  converted  into  sidereal  time 
gives,  for  the  three  observations. 


0,  =-  3*  29"*  1', 


<?„'=:=  21*  48"' 17', 


<?o"  =  21*  18"  55', 


which  arc  the  right  ascensions  of  the  geocentric  zenith,  of  which  <p' 
is  in  each  case  the  declination.  From  these  we  derive  the  longitude 
and  latitude  of  the  zenith  for  each  observation,  namely. 


I,  =      60°  33'.9, 
i„=  +  22    25.0, 


/;  =    347°    0'.4, 
6;=z  +  50    15.8, 


C  =    342°  59'.2, 
i;'=  +  53    41.6. 


Then,  by  means  of  equations  (4),  we  obtain 


aO„  =  — 18".92,  A©'  =  —  36".94,  aQ"  =  -  25".76, 

0.0001084,  A  log  i?„'  =  —  0.0002201. 


Alogi?, 


log. 
A  log  i?;'  =  — 0.0002796. 


For  the  reduction  of  time,  we  have  the  values  +  0M5,  +  0'.28,  and 
+  0'.34,  which  are  so  small  that  they  may  be  neglected. 


266 


THKOIiF/nCAT.    ASTRONOMY 


Filially,  tlio  longitiuk's  of  (M)tli  tho  sun  and  planet  are  reduced  to 
the  mean  ec^uinox  ol'  1863.0  by  applying  the  corrections 


50".95, 


-  51".o2, 


—  52'M4 ; 


and  the  latitudes  of  the  planet  an;  reduced  to  the  eclij)tie  of  the  same 
date  by  applying  the  corrections  —  0".15,  — O'M  1,  and  — 0".14, 
rcspectivelv. 

C'olleeting  together  and  applying  the  several  corrections  thus  ol)- 
tained  for  the  places  of  the  sun  and  of  the  planet,  reducing  the  un- 
corrected times  of  observation  to  the  meridian  of  Washington,  and 
expressing  them  in  days  from  the  beginning  of  the  year,  we  have  the 
following  tlata : — 

/o  =257.68079,  X  =  17°  46' 28".17,  /?  = -f- 3°    8' 43".r)l, 

to'  =  264.42r)70,  X'  =-.  16    40  2")  .19,  ,S'  =       2    52  27  .62, 

C'  =  271. .38625,  A"  =.  15    15  44.03,  ft"  = -{- 2    32  42.98, 

O   =172°    0'32".23,  logie  =0.0021056, 

0'  =178    35  48  .74,  log  A"  =0.0011656, 

O"  =  185    25  36  .90,  log  It"  =  0.0002378. 

The  numerical  values  of  the  several  corrections  to  be  applied  to 
the  data  furnished  by  observation  and  by  tiie  solar  tables  should  be 
checked  by  duplicate  calculation,  since  an  error  in  any  of  these  re- 
ductions will  not  be  indicated  until  after  the  entire  calculation  of  the 
elements  has  been  eifectcd. 

By  means  of  the  equations 


N-. 


tan  w'  = 


E'R"  sin  (Q"—Q') 
BR"  sill  {Q"—Qy 

tan  fi' 
sin  (A'  —  ©0' 


N"  =  -, 


ER's'iii(Q'—0) 


tan  ^' 


RR"^n  (©"—  ©)' 
tan  U'  —  ©') 


cosw 


we  obtain 

log  iV=  9.7087449,  log  N"  =  9.6950091, 

V  =  161°  42'  13".16, 

log  (R'  sin  4')  =  9.4980010,  log  {R'  cos  V)  =  9.9786355„. 

The  quadrant  in  which  oj/'  must  be  taken  is  determined  by  the  con- 
ditions that  i]/'  must  be  less  than  180°,  and  that  cos 4-'  and  cos(^'—  O') 
must  have  the  same  sign.     Then  froui 


NUMERICAL   EXAMPLE. 


267 


tan  /Hin  (},  (A"  -^X)-K)-=  ^''^J^'±^,  sec  ]  U"  -  X), 
^  -  '      2  cos ,  J  cos  (i         * 

tun/co.(i  (A"  +  A)  -  AT)  ..  ^'",!,'^'^|^  cosec^  U"-  A); 


tan  /9o  =  si"  C-^'  —  -K")  tan  /, 
,       EainiQ  ~-K) 


c  = 


__  win  dy  —  ,?„) 
cos  /J,,  tun  / ' 

Ji' mi  (Q'—K) 


(I 


Ji"  mi  (Q"—K) 


/=- 


sec  ,y 


m\{X"~;^y 


h^'- 


Rirmi(0"—Q) 
«„.sin(/."  — Aj    ' 


wo  conijinte  A',  /,  ,9,„  a^,  b,  c,  d,/,  and  //.  Tlio  anj^lc  /  nin.st  be  less 
tliiin  90°,  and  the  value  of  /%  nnist  1k'  dotcrniincd  with  tlui  jfreatest 
j)()ssil)l('  accuracy,  .since  on  tlii^  the  accuracy  of  the  resulting  elements 
principally  depends.     Thus  we  obtain 

K  ==  4°  47'  2}>".48,  log  tan  /  =  9.3884640, 

/9o  =  2°  52'  o!)"^  \l  log  «o  =:  0.801  J5583„, 

log  b  ^  2.54oG342„,  log  c  ^  2.2:}280oO^, 

logrfr=  1.2437914,  log/ =  1.3587437,.,         log/i -=  3.9247G91. 


The  forniuhe 


,.        sin  a" -A')    ,    .i?"sin(A"-Q") 

M,    =  -. — rrr, ;t  +J  


M, 


sin  (A"  — A) 
sin  (y  —  A) 
sin  (A"  —  A) 
hmii(X"—K) 
d 


/ 


d 
RsinU  —  Q) 


m: = 


hmi{k—K) 


give 


log  il/,"==  9.0690383, 
logiJ/;'==0.7306025„. 


log  i¥,  =  9.8940712, 
log  iV,=j:  1.9404111, 

The  quantities  thus  far  obtained  remain  unchanged  in  the  suc- 
cessive approximations  to  the  values  of  P  and  Q. 
For  the  first  hypothesis,  from 


P=-, 

T 

b  +  Pd 
1  -I-  P' 


Q  =  "", 


"'0  —  "o        "> 

Tjo  sin  C  =  i?'  3in  V, 

ijo  cos  C  =  /jfl  —  P'  cos  v. 


^o=-iCo§, 


/„ 


««n 


FT!" 


ij,ie"sin^4 


268 

THEORETICAL 

ASTKOXOMY. 

we  obtain 

lOgT     =r 

:  9.0782249, 

]ogr"=:=.9.0()45575, 

logP=- 

9.98()882(], 

log  Q  ^  8.1427824, 

log^„  = 

:  2.22985(i7„, 

log /'a  -.0.0704470, 

log^o  - 

.0.07i(i091, 

loglJo   :r.:.  0.;W2()'.)25, 

C  = 

:  8°  24'  49".74, 

log?Ho=1-2449l;>(J. 

The  (quadrant  in  whicli  !^  must  bo  situated  is  determined  by  tJie  eoii- 
dition  that  r^^  shali  liave  the  same  sijj;n  as  /„. 

The  \'ahu>  of  z'  must  now  be  found  by  trial  from  tlie  C(|Uation 

sin  iz'  —  ?)  =r  7Ho  sin*  z\ 

Table  XII.  shows  that  of  the  four  roots  of  this  equation  one  exceeds 
180°,  and  is  therefore  excluded  by  the  condition  that  sin,-'  must  be 
positive,  and  that  tAvo  of  tliese  roots  give  z'  greater  tlian  180^  —■4'') 
and  are  excluded  )w  the  condition  thi'.t  z'  must  be  less  than  180°—  ^'. 
The  remaining  root  is  that  which  belongs  to  the  orbit  of  the  planet, 
and  it  is  shown  to  be  approximately  10°  40' ;  but  the  correct  value 
is  found  from  the  last  ecpuition  by  a  few  trials  to  be 

z'  ^  9°  r  22".96. 

The  root  which  corresponds  to  the  orbit  of  the  earth  is  18°  20'  41", 
and  differs  very  little  from  180°  —  t^'. 
Next,  from 


/  =: 


J?\sin4' 

sin  z'   ' 


R'm\(z'-\-^') 


sms 


cos/S', 


n 


i^p(i +  ,'?.).    .."^-=«p, 


Ave  derive 


logr'=.0..']025r,72, 
log  n==^- 9.70(11229, 
log/,  =:=.  0.0254823, 


hgp'  ^0.0123991, 
logJi"=^9.()92*555, 
log/>"^  0.0028859. 


The  values  of  the  curtate  distances  having  thus  been  found,  the 
heliocentric  places  for  tlie  three  observations  are  now  computed  from 


NrJrERICAL   EXAMPLE. 


269 


r  cos/)  cos  (/  —  O) 
r  cos/>  ^in  (J —  ©) 
rnmb 
?-'co.s6'cos(r— 0') 


=  /ocos(A  —  O)  —  E, 
=  P  sin  (^~  Q), 
=  /)  tan  /5 ; 

:/)'cos(/'— ©')  — -R', 


r'  COS  //  sin  (7'  -  -  ©')      :--.  //  sin  (/'  -  © '), 


r  sin  l> 


-^  //  tan  fi' ; 


1-"  t-os  b"  COS  (Z"  -  -  ©")  =  f>"  cos  (A"  —  ©")  - 
r"  COS  6"  sin  (/"  -  ©")  =  />"  sin  (/'  -  ©"), 


B", 


r"  sin  h" 


■  p"tvin(i", 


wiuoli  give 


/  ::.  5°  14'  39".a3, 
/'  = :  7  45  11  .28, 
r'-^lO    21  34  .57, 


log  tan  b  --=  8.4615572, 
log  tan  6'  :r^  8.41075.55, 
logtan6"  =  8.:i497i)1l, 


logr  =r  0.:{040094, 
log  r'  ==  0.;5025»>73, 
logr"  ^0.3011010. 


The  agreement  of  the  value  of  logr'  thus  ol)tained  with  tliat  already 
tbund,  is  a  proof  of  part  of  the  calculation.     Then,  from 


tantsin(.Ur'-t-0— S^) 


tan  h"  4-  tan  b 

iTcos^U"  — 0' 
tan  b"  —  tan  b 


tau  V  - 
wo  get 


^  ,  /  111    ,      i\  ^  •»  mil  u     —  tunc/ 


cos  I 


cos^ 


cos  I 


SI  =  207°  2'  38".16,  i  =  4°  27'  23".84, 

u .    158°  8'  25".78,  «'  =  160°  39'  18".13,  «"  --  163°  16'  4".42. 


The  equation 


tau  b'  ■-=  tan  i  sin  (I'  —  SI) 


^■ivos  log  tan/>'  8.4107514,  which  diifers  0.0000041  from  the  value 
already  found  directly  from  //.  This  ditlerence,  however,  amounts 
to  only  0".05  in  the  value  of  the  heliooeutric  latitude,  and  is  due  to 
errors  of  calculation.     If  we  com{)ute  w  and  n"  from  the  ecpiatious 


n  = 


r'r"mi(u"-u') 
rr"Hin(,i«''— <0' 


n"  = 


rr'  sin{(('  —  n) 
rr"  sin  (u"  —  it)' 


the  results  should  agree  with  the  values  of  these  (piantities  ]n'eviously 
fomputed  directly  from  P  and  Q.  Using  the  values  of  «,  w',  and 
1'"  just  found,  we  obtain 


log  «  =9.7061158, 


log  ri":=  9.6924683, 


270 


THEORETICAL   ASTRONOMY. 


which  (liifor  in  the  last  decinial  i)laf'e.s  from  the  values  used  in  findiii"- 
[)  and  ft".     According  to  the  equations 

d  log  H  =:  —  21 .055  cot  ( it"—  ?t')  du', 
d  log  n"  =  4-  21.055  cot  («'  —  ?<)  da', 

the  differences  of  log »  and  log/*"  being  expressed  in  units  of  the 
seventh  decimal  place,  the  correction  to  «/  ne(;essaiy  to  make  the  two 
values  of  log/*  agree  is  — 0".15;  but  for  the  agreement  of  the  two 
values  of  log??",  u'  must  be  diminished  by  0".26,  so  that  it  appears 
that  this  proof  is  not  com])lete,  although  near  enough  for  the  first 
approximation.  It  should  l)c  observed,  however,  that  a  great  circle 
passing  tlirough  the  extreme  observed  places  of  the  i)lanet  passes 
very  nearly  through  the  third  place  of  the  sun,  and  hence  the  values 
of  p  and  f>"  as  determined  by  means  of  the  last  two  of  e(inations  (18) 
are  somewhat  uncertain.  In  this  case  it  would  be  advisable  to  com- 
pute o  and  fi",  as  soon  as  ft'  has  been  found,  by  means  of  the  equa- 
tions (22)  and  (23).     Thus,  from  these  equations  we  obtain 


log 

i"  = 

0.0254918, 

log/."  = 

0.00288 

74, 

and  hence 

I 

=   5° 

14 

40".  05, 

log  tan  h  - 

=  8.401 

5019, 

log 

r  = 

0.3041042, 

I" 

=  10 

21 

34 

.li), 

log  tan  b"  - 

=  8.3497919, 

log) 

,11 

0.30]  1UI7, 

9,^ 

=  20 

r  2'  32".97, 

i 

=  4° 

27'  25". 

13, 

U- 

=  158^ 

'  8' 

31" 

.47, 

u'  =  160° 

39'  23" 

31, 

«"  = 

163 

°  16'  9". 22. 

The  value  of  log  tan  b'  derived  from  /'  and  these  values  of  Q>  and  i, 

is  8.4107555,  agreeing  exactly  with   that  derived  from  ft'  directly. 

The  values  of  n  and  n"  given  by  these  last  results  for  u,  u'  and  n'', 

are 

log  //  =  9.7001144,  log  ii"  ==  9.0924040 ; 

and  this  proof  will  be  complete  if  we  apply  the  correction  du'^=—0".lS 
to  the  value  of  u',  so  that  we  have 

h"  ~  «'  =  2°  36'  46".09,  n'  —  u  =  2°  30'  51".66. 

The  results  which  have  thus  been  obtained  enable  us  to  proceeil  to 
a  second  approximation  to  the  correct  valuers  of  P  and  Q,  and  we 
may  also  corn.'ct  the  times  of  observation  for  the  time  of  aberration 
by  means  of  the  formulae 

t^t^—Cp  sec ,?,  f  =  f,;  ~  Cp'  sec  f/,  t"  =  t," -  Cp"  sec  jf', 

\v\u'f*»n  log  C-=  7.760523,  cx^eantA  m  partt<  of  a  day.   Thus  we  get 

4  ^--  257,67467,  if  ---.  ^.#074^  <"  =  27 1 .38044, 


NUMERICAL   EXAMPLE. 


271 


and  hence 

Wr  =  9.0782331, 


log  r'  .=  9.8724848, 


loL'r"==  9.0045692. 


Thou,  to  find  tlie  ratios  denoted  by  s  and  s",  wc  have 


m 


sin  ycofiG  =  sin  -S  {u"  —  u'), 

sin  y  sin  G  =  cos  A  {u"  —  u')  cos  2/, 

cos  /-  =  cos  tI  («"  —  v!)  sin  2/ ; 

tan;^"..-^^, 

sin  •/'  cos  G"  ---  sin  h  (u'  —  u), 

sin  r"  sin  G"  =  cos  l  iu'  —  u)  cos  2/', 

cos  /'  =  COS  A  (,n'  —  u)  *^hi  2/f" ; 

T*  .   sin' .')' 

•^         cos  y ' 

sin^V: 
•^   ~  cos>"  ' 


(r'+/-")'cosV' 


,"2 


m"  = 


(,._|-,.')3C0SV"' 

from  wliich  we  obtain 

X  =■  44°  57'    6".00, 
y==    1    IM  85  .90, 
log?»i  =  6.84H2ll4, 
logj  =  6.1^8185, 


X"  --=  44°  56'  57".50, 
/'  =    1    15  40  .69, 
log  Hi"  =.  6.8168548, 
log/'  =  6.0834230. 


From  the.se,  by  means  of  the  equations 

m 


m 


using  Tables  XI 11.  an«l  XIV.,  we  compute  s  and  s".     First,  In  the 
ca^o  of  s,  we  assume 


m 


:  0.0002675, 


and,  with  this  a«  the  argument,  Tablf  X  III.  trive.s  log  s'-  ^  0.0002581. 
Hence  we  obtain  j;' ==  0.0OfX>92,  ami,  with  this  as  the  argument, 
Table  XIV.  gives  f  0.00<J</<XJ01 ;  and,  theivfore,  it  appears  that  a 
repetition  of  the  ealculation  i,s  unuece»«iry.     Thus  we  obtain 

logs  ^mm-m,  log./' =^0.0001200. 

VVii-  r)  the  intervals  -jm  maA\,  it  is  not  necessary  to  u«c  the  formula) 


272 


THEORETICAL   ASTRONOMY. 


in  tlic  coin]>lote  form  here  givoii,  since  tliese  ratios  may  then  l)e  fomid 
by  a  simpler  process,  as  will  appear  in  the  secj^uel.     Then,  from 


r 
rr" 


.." ' 


we  find 


^  ~"  6'«"  '  r?'  cos  h  (u"  —  u'fcos  \  [vT—  u)  cos  ^  (u'  —  n)' 
log  P=  9.9863451,  log  Q  =  8.1431841, 


wit!\  which  the  second  approxiination  may  be  completed.  AVc  now 
compute  (',„  /•„,  /,„  z',  etc.  })recisely  as  in  the  first  api)roximation;  but 
we  shall  prefer,  for  the  reason  already  stated,  the  values  of  />  and  //' 
computed  by  means  of  the  equations  (22)  and  (23)  instead  of  those 
obtained  from  the  last  two  of  the  formukc  (18).  The  results  thus 
derived  are  as  follows: — 

log  r,  r=.  2.2298499„,  log  k,  =--  0.0714280, 

log  [„  ^_  0.07 19540,  log  r/„  ^  0.3332233, 

C  :^  8°  24'  12".48,  logw„=  1.2447277, 

/  =  9°  0'  30".84, 
log 7-'  =.  0.3032587,  log/*'  =  0.0137G21, 

log  )i  ^-.  9.70(!1153,  log  )>"--=.-.  9.r)924(i04, 

\ogi>  =.  0.02G9143,  log//' =  0.0041748, 

I  =  5°  1.5'  57".26,  log  tan  b  =  8.4622524,  log  r  =  0.3048308, 
^' =  7  46  2.76,  log  tan  6'  =8.4114276,  log /•' =  0.3032.W, 
I"  =  10    22     0  .91,         log  tan  b"  =  8.3504332,         log  r"  =  0.3017481, 

Q,  =  207°  0'  0".72,  i  =  4°  28'  35".20, 

u  =  158°  12'  19".54,         u'  =  160°  42'  4o".82,         u"  =  163°  19'  7".14. 

The  agreement  of  the  two  values  of  log?''  is  complete,  and  the  value 
of  log  tan  b'  computetl  from 

tan  b'      tan  I  sin  d'  —  Sl)> 

is  log  tan  6'==  8.4114279.  agreeinir  with  the  result  derived  directly 
from  /}'.     The  values  of  n  and  n"  obtained  from  the  equations  (54i 

are 

log  a  -.=:  9.7061156,  log  n"  =  9.6924^..:^ 

which  agree  with  the  valiu^*  already  used  in  computing  «  and  //',  and 
the  proof  jf  the  calculation  is  ••omplcto.     We  hmv^,  tliinvtbre, 

u"—  u'  =  2°  36'  21".32,    «'—  u  =  2°  30'  26".a?\    n"—  «  =  5°  G'  47".60. 

From  these  values  of  it" —  u'  and  u'  —  **>  we  obtain 
log  s  =  O.«>001284,  log  a"  =  0.0001193, 


NUMEEICAL   EXAMPLE. 


273 


aud,  recomputing  P  and  ^,  we  get 

log  P  :=.  9.98G3452,  log  §  =  8.1431359, 

which  differ  so  little  from  the  preceding  values  of  these  quantities 
that  another  approximation  is  unnecessary.    We  may,  therefore,  from 
the  results  already  derived,  complete  the  determination  of  the  elements 
of  the  orbit. 
The  equations 

tan/  =  -^-, 

sin  y'  cos  Q'  =  sii.  h  (tt"  —  u), 
sin  y'  sin  G'  =  cos  \  (u"  —  u)  cos  2/', 
cos  /  =  cos  J  Qii"  —  u)  sin  2/', 


m'  = 


(r -\- r"  f  cos' y 


.3„" 


sin'  A/ 

cos  J''  ' 


give 

■/  =  44°  53'  53".25,         /  =  2°  33'  52".97,         log  tan  G'  =  8.9011435, 
log  m'  =  6.9332999,  log/  =  6.7001345. 

From  these,  by  means  of  the  formulce 


m 


+/  + 


*  —  s'»  —J  ' 


and  Tables  XIII.  and  XIV.,  we  obtain 


Then  from 


we  get 


log  8''  =  0.0009908, 


log  .7;'  =  6.54941 16. 


-( 


s'rr"  sin  (u"  —  u) 


logi>  =  0.3691818. 
The  values  of  logp  given  by 


-( 


s/r"  sin  (u"  —  u') 


y_l8"rr'mUu'—n)Y 


are  0.3691824  aud  0.3691814,  the  mean  of  which  agrees  with  the 
result  obtained  from  lo"  —  u,  and  the  differences  between  the  separate 
results  are  so  small  that  the  approximation  to  P  and  Q  is  sufficieut. 
The  equations 


a  cos  y  = 

003^9  = 


sin^  (u"  —  u) 
sin^  {E"—E) 

_P 

a  cos  ^' 


Vrr", 


•>j4i 


271 


TIIEORETIOAL   A.STnOXOMY. 


give 


1  iE"  —  E) .-- 1°  4'  42".!»03,  lofx  (n  cos  <p)  =  0.3770315, 

log  cos  9?  =  9.9921503. 
Next,  from 

e  sin  (w  —  I  iv!'  -f-  «))  -^ ^'  -~-  tun  (/', 

cos  /  V  rr" 


e  cos  (w  —  ^  (u"  +  «)) 


_  ?' 

cos /l/ it" 


sec  ^  06"  —  m), 


wo  obtain 


w  =.  190°  15'  39".57, 
<P  =    10    51  39  .62, 


log  c  =  log  sin  9?  :==  9.2751434, 

ri.^  «.  +  f^  = :  37°  15'  40".29. 


This  value  of  tp  gives  log  cos^  =  9.9921501,  agreeing  with  the  result 
already  found. 

To  lind  a  and  /^,  wo  have 


a 


P 

cos'''  <p 


ti- 


the value  of  /•  expressed  in  seconds  of  arc  being  log/;=  3.5500066, 
from  which  the  results  are 

log  a  =  0.3848816,  log  /x  =-  2.9720842. 

The  true  anomalies  are  given  by 

V  ==^t  —  w,  v'  =  ll'  —  w,  v"  =^  'ill'  —  w, 

according  to  Avhich  we  have 

V  =  327°  56'  39".97,  v'  =^=  330°  27'  6".25,  v"  =  333°  3'  27".57. 

U  we  computer  r,  v' ,  and  /■"  from  these  values  by  means  of  the  polar 
equation  of  the  ellipse,  we  get 


losr /•=- 0.3048367, 


logr'  -■=  0.3032586,  log  /•"  =  0.3017481, 


and  the  agreement  of  these  results  with  those  derived  directly  from 
(),  p' ,  and  (i''  is  a  further  proof  of  the  calculation. 
The  equations 

ia\i\E  =tan(45°  — ^9')tan^v, 

tan  \E'  =  tan  (45°  —  if)  tan  \v', 

tan  Ie"  =-  tan  (45°  —  y)  tan  ^v" 
give 

t:  =  333°  17'  28".|8,       ^'  =:  835°  24'  38".00,      E"  =-  337°  36'  19".78. 


.NTTMKinrA r,  kxa m it,k. 


275 


The  vmIuc  of  \{E"  -  E)  tlius  obtained  differs  only  0".00;}  from  that 
coiiipuft'd  dirt'clly  i'ronj  .r' , 

Finally,  I'or  the  nu'an  anomalies  we  liave 

II      A'— e  sin  7s;  M'  -^r- E'  —  e  ^\\\  E\  M"   -- E"  —  e  :<m E" , 

from  wliieli  wc  f^et 

M  ---  .338°  8'  3r)".71,       M'  -=  339°  54'  W.Cyl,       M"  --=-.  341°  43'  r.".97  ; 

and  if  ^1/^,  denotes  the  mean  anomaly  for  the  date  7'-   1803  Sept.  21.5 
Wusiungton  mean  time,  from  the  formnla; 

M^M  —,i(i  —  T) 
^M'  —lilt'—T) 
=  M"  —  //  (/"  -  T), 

we  obtain  the  three  values  339°  55'  25".97,  339°  55'  25".96,  and 
339°  55'  25".9(),  the  mean  of  whieh  j^ives 

il/-„  =  339°  55'  25".96. 

The  ac;reement  of  the  three  resnlts  for  J/y  is  a  final  proof  of  the 
iiccaracy  of  the  entire  calenlation  of  the  elements. 

Collectin<ij  toi^ether  the  separate  results  obtained,  we  have  the  fol- 
lowing elements : 

E])oeli  --- 18()3  Sept.  21.5  Washington  mean  time. 
j|/-=-.339°  55'  25".96 
r  =    37    15  40  .2!) 
ft  ^  207      0     0  .72 
i=     4    28  35  .20. 
^  =    10    51  39  .(32 
log  a  =  0.3848816 
log//:^2.972()H42 
/x  =  939".04022. 

If  we  rom])ute  the  geoeentrie  right  ascension  and  declination  of 
the  ])lanet  directly  from  these  elements  for  the  dates  of  tiic  ol)serva- 
lioiis,  as  corre(!ted  for  the  time  of  aberration,  and  then  reduce  the 
oliscrvations  to  the  eentr(i  of  the  earth  by  a|)|)lying  the  (rorreetions 
liir  parallax,  the  comparison  of  the  results  thus  obtained  will  show 
Ik'w  closely  tlu'  elements  represent  the  places  on  which  they  are 
hiiM'd.  Thus,  we  comi)ute  first  the  auxiliary  constants  for  the  equator, 
using  the  mean  obliquity  of  the  ecliptic, 

e  =  23°  27'  24".96, 


J'^cliptic  and  Mean 
Equinox  18(33.0. 


27G 


TIIEOIIETICAL   ASTRONOMY. 


and  tlie  following  expressions  for  the  heliocentric  co-ordinates  of  the 
planet  arc  ohtained  : 

x^r  [0.0007272]  sin  (206°  5,5'  46".05  +  n), 
y=r[i).i)744()00]  will  (20(5  12  42  .70  +  »), 
2  =i=r  [0.5240530]  sill  (212    30  14  .02  +  u). 

The  niunhors  enclosed  in  the  brackets  are  the  logarithms  of  sin  o, 
sin 6,  and  m\c,  respectively;  and  these  c<inations  give  the  co-ordinates 
referred  to  the  mean  eqninox  and  ecjuator  of  18G3.0. 

The  p]a(;es  of  the  sun  for  the  corre(!ted  times  of  observation,  and 
referred  to  the  mean  equinox  of  18G3.0,  arc 


True  Loiif^itude. 

Latitude. 

Log  R. 

172°    0'  29".5 

-  0".07 

0.0022146, 

178    36    4  .5 

-1-0  .77 

0.0013864, 

185    25  42  .0 

-t-0  .67 

0.0005182. 

If  we  compute  froiti  these  values,  by  means  of  the  equations  (104)i, 
the  co-ordinates  of  the  sun,  and  combine  them  with  the  corresponding 
heliocentric  co-ortlinates  of  the  planet,  we  obtain  the  following  geo- 
centric places  of  the  planet : 


a  r=15°  10'29".06, 
a'  =  14  15  0  .22, 
a"  =  13      3  40  .47, 


<J  =:  +  9°  53'  16".72, 
5'  =  9  12  51  .29, 
,3"  ^  +  8    21  54  .46, 


logj  =0.02726, 
log  J'  =  0.01410, 
log  J"  =  0.00433. 


To  reduce  these  places  to  the  apparent  equinox  of  the  date  of  obser- 
vation, the  corrections 

4-  48".14,  -I-  48".54,  4-  48".91, 

must  be  applied  to  the  right  ascensions,  respectively,  and 

-fl8".55,  -fl8".92,  -fl9".31, 

to  the  declinations.     Thus  we  obtain  : 


Wasliington  M.  T. 

1863  Sept.  14.67467 
21.41976 
28.38044 


Conip.  a. 

1*  0"  45M5 
0  57  3.25 
0  52    18.56 


Comp.  6. 
+  9°  53'  35".3, 
9    13  10  .2, 
-1-8    22  13  .8. 


The  corrections  to  be  applied  to  the  respective  observations,  in  order 
to  reduce  them  to  the  centre  of  the  earth,  are  -|-  0'.24,  —  O'.Sl,  —  0'.34 
in  right  ascension,  and  +  4".5,  -f-  4".8,  -f-  5".l  in  declination,  so 
that  we  have,  for  the  same  dates. 


NUMERICAL   EXAMPLE. 


277 


Observed  a. 

1*  0"  45M5 
0  57  3.26 
0  52    18.56 


Observed  <'. 
4-  9°  53'  35".3, 
9    13  10  .3, 

+  8    22  13  .8. 


The  comparison  of  those  with  the  computed  vahi(\s  shows  that  the 
cxtronie  phices  are  exactly  represented,  while  the  diirerenee  in  the 
middle  place  amounts  to  only  O^.O!  in  riji^ht  ascension,  and  to  O'M 
ill  declination.  It  ajjjjoars,  therefore,  that  the  ohservations  are  com- 
pletely satisfied  V)y  the  elements  obtained,  and  that  the  preliminary 
corrections  for  aberration  and  parallax,  as  determined  by  the  ecjua- 
tioiis  (1)  and  (4),  have  been  correctly  computed. 

It  cannot  be  expected  that  a  system  of  elements  derived  from  ob- 
servations includinj;  an  interval  of  only  fonrteen  days,  will  be  so 
exact  as  the  results  which  are  obtained  from  a  series  of  observations 
or  from  those  includint?  a  much  longer  interval  of  time;  and  although 
the  elements  which  have  been  derived  completely  represent  the  data, 
yet,  on  account  of  the  smallness  of  /9'  — /9„,  this  ditte'rence  being  only 
31". 893,  the  slight  errors  of  observation  have  considerable  influence 
ill  the  final  results. 

When  approximate  elements  are  already  known,  so  that  the  cor- 
rection for  parallax  may  be  applied  directly  to  the  observations,  in 
order  to  take  into  account  the  latitude  of  the  sun,  the  observed  places 
of  the  body  must  be  reduced,  by  means  of  e(piation  (6),  to  the  point 
ill  which  a  perpendicular  let  fall  from  the  centre  of  the  earth  to  the 
plane  of  the  ecliptic  cuts  that  plane.  The  times  of  observation  must 
also  be  corrected  for  the  time  of  aberration,  and  the  corresponding 
places  of  both  the  planet  and  the  sun  must  be  reduced  to  the  ecliptic 
and  mean  equinox  of  a  fixed  epoch ;  and  further,  the  reduction  to 
the  fixed  ecliptic  should  precede  the  aiiplication  of  ecpiation  (0). 

If  the  intervals  between  the  times  of  observation  are  considtu-able, 
it  may  become  necessary  to  make  three  or  more  a[)proximations  to  the 
values  of  P  and  Q,  and  in  this  case  the  equations  (82)  may  be  ap[)lied. 
But  when  approximate  elements  are  already  known,  it  will  l)c  advan- 
tageous to  compute  the  first  assumed  values  of  P  and  Q  directly 
from  these  elements  by  means  of  the  equations  (44)  or  by  means  of 
(48)  and  (51) ;  and  the  ratios  s  and  h"  may  be  found  directly  from  the 
equations  (46).  In  the  case  of  very  eccentric  orbits  this  is  indispen- 
sable, if  it  be  dcsii'cd  to  avoid  prolixity  in  the  numerical  calculation, 
since  otherwise  the  successive  approximations  to  P  and  (2  will  slowly 
apjiroach  the  limits  required. 


278 


TIIKOUKTUAI-    AHTUONOMY. 


Tlio  various  niodifimtioiis  of  tlu^  rornmlii'  lor  certain  spofial  rasos, 
as  well  as  tlu!  (onmilii'  wlilcli  iiiiist  ix'  used  in  tiic  (■as(!  ol"  j)aral)()ii(; 
and  Iiy|)('ri)()lif  ()rl)its,  and  of  tlio-ic  tliifcrini:;  i)Ut  litth;  Ironi  tho 
parabola,  liavo  hccn  given  in  a  ibriu  sneli  tliut  they  rctjuiru  no  fur- 
ther illustration. 

94.  In  tho  (leterniination  of  an  unknown  orbit,  if  the  intervals  are 
(,'onsiderably  nne(|Ual,  it  will  be  advantageous  to  correet  the  lirst 
assnnu'd  value  of  J*  before  eonipleting  the  fust  aj)proxiniation  in  the 
manner  already  illustrated.     The  assumption  of 

Q  =.  rr" 

is  correet  to  t(  rms  of  the  fourth  order  with  rcspeet  to  tho  time,  and 
for  the  same  degree  of  approximation  to  P  we  must,  uceording  to 
erpiatiou  ('2(S).„  use  the  ex])ression 


T 
T 


which  becomes  equal  to  - —  only  when  the  intervals  arc  c(j[ual.     Tlu.' 
first  assumed  values 


T 
T 


Q^rr", 


furnish,  Mith  very  little  labor,  an  a])proximatc  value  of  r' ;  and  then, 
with  the  values  of  7*  and  Q,  derived  from 


t"  /  t'  —  t"'^  \ 


(98) 


the  entire  calculation  should  be  completed  precisely  as  in  the  example 
given.     Thus,  in  this  example,  the  lirst  assumed  values  give 

log  r' -.  0.30257, 

and,  rccomjniting  P  by  means  of  the  first  of  these  equations,  we  get 

log  P=  9.9863404,  log  q  ^  8.1427822, 

Avith  which,  if  the  first  approximation  to  the  elements  be  com])lete(l, 
the  results  will  differ  but  little  from  those  obtained,  without  this  cor- 
rection, from  the  second  hyj)othesis.  If  the  times  had  been  already 
corrected  for  the  time  of  aberration,  the  agreement  would  bo  still 
closer. 

The  comparison  of  equations  (46)  with  (20)3  gives,  to  terms  of  the 
fourth  order, 


NUMERICAL   EXAMPLE. 


279 


«=l  +  i  .p         «'-  1  +  r;,..         «"  =  l  +  i 


.»i 


ami,  if  tli((  intervals  jirc  ('(jiial,  this  valuu  ul'  «'  is  correct  to  terms  of 
the  lit'tli  order.     8iiiee 

loge  s  =  logo  (1  +  (.V  -  1 ))  =  «  -  1  -  A  0^  -  1)"  +  Ac, 

wc  have,  neglecting  terms  of  the  fourth  order, 


(.91)) 


ill  which  log  JA„  ---  8.85!)().')o0.     AVc  have,  also,  to  the  same  degree  of 
:i[)|ti'<iximation, 

log  ."  =  >.--.  (100) 


ioB-=f  v:. 


For  the  values 


lo^,'  T  =  9.0782331,  log  t'  ^^  9.3724848,  log  r"  =  9.0045092, 

log /•' =-^  0.3032-387, 
tiic^e  formulsD  give 

log  8  =  0.0001 277,  log  s'  :-=:  0.00049o3,  log  s"  =  0.0001 199, 

whicli  diller  but  little  from  the  correct  values  0.0(M)1281,  0.0001954, 

and  0.0001193  previously  obtained. 

Since 

secV=-l  +  6  .si^^y  +  &c., 

the  second  of  equations  (G5)  gives 

Substituting  this  value  in  the  first  of  equations  (GG),  we  get 

If  we  neglect  terms  of  the  fourth  order  with  respect  to  the  time,  it 
will  be  sufficien:.  lu  'bis  equation  to  put  y'  ---  f,  according  to  (71),  and 
hi'iice  we  have 

and,  since  .s'  —  1  is  of  the  second  order  with  respect  to  r',  we  have, 
to  terms  of  the  fourth  order, 

8'»(8'-l)=l0g,s'. 


IMAGE  EVALUATION 
TEST  TARGET  (MT-3) 


II  I.I 
11.25 


Photographic 

Sdoices 

Corporation 


23  WIST  MAIN  STREIT 

WEBSTfR  N.Y.  I4SM 

(716)  S73-4503 


rtV 


iV 


•1>' 


s> 


'^ 


%^ 


4^> 


I/a 


i 


280 
Therefore, 


THEORETICAL  ASTRONOMY. 


^-s^-t'oj—:^ 


(101) 


whicli,  when  the  intervals  are  small,  may  be  used  to  find  s'  from  r 
and  r".     In  the  same  manner,  we  obtain 


t"' 


log  s  =  t^,  -(7q.-7Tya.  log  s"  =  ti,  (^"Ty.- 


(102) 


For  logarithmic  calculation,  when  addition  and  subtnu'tion  loga- 
rithms are  not  used,  it  is  more  convenient  to  introduce  the  auxiliary 
angles  ^,  x't  *^"*^  X">  ^Y  w^***"'*  of  which  these  formula!  become 

,  .,  T'''cos*y  ,  ,  .,  t'^cos'/  ,  ,,  .,  t"»cosV'  /<««% 
log 8  =  iX, -~ ^ ,,  \     log «'  =  iX,  — -^  ^  ,     log «"  =  ^^, ^  ^  ,  (103) 

in  which  log  ^A„-=  0.7627230.     For  the   ^M'st   approximation  those 

e(juations  will  l)e  sufticlent,  even  when  the  intervals  are  considerable, 

to  determine  the  values  of  «  and  h"  required  in  correcting  P  and  Q. 

The  values  of  r,  r',  r",  and  /•"  above  given,  in  connection  with 


logr=::  0.3048308, 


log  r"  =  0.3017481, 


give 
logs  ==0.0001284, 


log  «'  =  0.0004951, 


logs"  =  0.0001193. 


These  results  for  log«  and  log«"  are  correct,  and  that  for  logs'  differs 
only  3  in  the  seventh  decimal  place  from  the  correct  value. 


ORBIT  FROM   FOUR  OBSERVATIONS. 


281 


CHAPTER  V. 

DETERMINATION  OF  THE  OUHIT  OF   A  HKAVKNI.Y  BODY  FROM   FOUR  OBflF.KVATIONS, 
OP  WHICH   THE  SECOND  AND  THIRD   MIST  BE  COMI'I^FIE. 

95.  The  formultc  given  in  the  preceding  chapter  are  not  sufficient 
to  (li'terniine  the  elements  of  the  orbit  of  a  heavenly  hody  when  its 
apparent  path  is  in  the  plane  of  the  ecli|)tic.  In  this  case,  however, 
the  position  of  the  plane  of  the  orbit  being  known,  only  four  elc- 
nioiits  remain  to  be  determined,  ai,d  four  observed  longitudes  will 
furnisii  the  necessary  equations.  There  is  no  instance  of  an  orbit 
whose  inclinr.tion  is  zero;  but,  although  no  such  case  may  wicur,  it  may 
happen  that  the  inclination  is  very  small,  and  that  the  elements 
derived  from  three  observations  will  on  this  acc'ount  be  uncertain, 
and  esjjeeially  so,  if  the  observations  are  not  very  exa<'t.  The  dilli- 
culty  thus  encountered  may  be  remedied  by  using  for  the  tlata  in  tiic 
determination  of  the  elements  one  or  more  additional  ol)scrvations, 
and  neglecting  those  latitudes  which  are  regarded  as  most  uncertain. 
The  formula;,  however,  are  most  convenient,  and  lead  most  exj)e- 
ditiously  to  a  knowledge  of  the  elements  of  an  orbit  wholly  unknown, 
when  they  are  made  to  depend  on  four  observations,  the  si-coiul  and 
third  of  which  must  be  complete ;  but  of  the  extreme  observations 
only  the  longitudes  arc  absolutely  required. 

The  preliminary  reductions  to  be  applied  to  the  data  are  derived 
precisely  as  explaincil  in  the  preceding  chapter,  preparatory  to  a  de- 
torniination  of  the  elements  of  the  ori)it  from  three  observations. 

Let  /,  r,  t",  i'"  be  the  times  of  observation,  r,  /•',  r",  r'"  the  radii- 
vcctores  of  the  body,  «,  u',  u" ,  u'"  the  correspomling  arguments  of 
the  latitude,  R,  li',  R",  R'"  the  distances  of  the  earth  from  the  sun, 
and  O,  O',  0",  ©'"  the  longitudes  of  the  sun  corresponding  to 
these  times.     Let  us  also  put 


and 


[//"]  =  r>r"'  8in  («'"  -  «'), 
[,•"/"]  =  r"r"' sin  (u"'  —  m"), 


n' 


\r"r"'^ 


n 


(1) 


282  TUKOIJKTICAIi    iV)?T«OXOMV. 

Then,  lu-cording  In  the  ('(luatioiis  ('))„  we  i^hall  have 


ny->'-f/."V"      0, 


iV 


Lot  /,  /',  /",  //"  bo  the  ohsorvod  lonirididcs,  ,'i,  ,V,  ,i",  ,i"'  tho  oh- 

served  latitudes  eorres|)oiidiii<f  to  tlie  tiiius  f,  f,  (",  /'",  respeetivcly, 

and  J,  J',  J",  J'"  the  tlistaiiees  ot' tlu'  IhxIv  iVoin  the  earth.     l''urtlitr, 

let 

J"'co8,i"'=://", 

and  for  the  lust  place  we  have 

ar"'  =  //"eo8/"'-7reosO"', 
f  =^  ,,'"  mx  a'"  —  ir  sin  ©'". 

Introdueiiij;  those  values  of  .)•'"  and  //'",  and  the  «'orrospondin<j;  values 
of  .r,  .!•',  .)'",  //,  y',  I/"  into  the  o(ination.s  (2),  they  heeouie 

0  =  n{p  cos ^  —  JtcosQ)  —  (//  cos /.'  —  7."  ri)i^Q') 

+  //"(/>"  cos  r  — /r  cos  o"), 

0  —  ?t  (/>  i^in  -I  —  Ji  sill  O)  —  (//  »iin  /.'  —  7»"  sin  O' ) 

+  «"(>/' sin  r  —  i2"  sin  0"), 
O  =  u'(//eos/l'  — i^'cosO')  — 0/'co8/"-  A'"eusO")  (3) 

+  „"'(//"eosr'-/?'"cosO"'), 
0  =  n'  (//  sin  A'  —  it"  sin  ©')  —  (f>"  sin  /•"  -  If"  sin  ©") 

-f  »'"(//"  siiU"'-7r  sin  0'"). 

If  we  niultijdy  the  first  of  those  eqnations  by  sin  ^,  and  the  second 
by  —cos/,  and  add  the  j)rodnets,  we  get 


0  =  nJi  sin  (X  -  Q)  —  {p'  sin  (;.'  —  ;.)  +  /?'  sin  (A  —  ©')) 
4-  n"  {p"  sin  ( /  '  — ;. )  +  n"  sin  (.;.  —  ©  ")) ; 


(4^ 


and  in  a  similar  niannei,  from  the  third  and  fonrth  equations,  wo 
find 

0  =  «'(//sin(r'  — A')--/rsin(>l"'— ©'))  (5) 

-  (//'  sin  (A'"-  ;")  —  li"  sin  (/"-  ©"))  -  n"'Ji"'  sin  (A'"-  0'" ). 

Whenever  the  values  of  n,  n',  ii'',  and  »'"  are  known,  or  may  he 
dotennined  in  functions  of  the  time  so  as  to  satisfy  the  conditions  of 
motion  in  a  conic  section,  these  e(|uations  become  distinct  or  inde- 
pendent of  each  other ;  and,  since  only  two  uidcnown  quantities  // 


OKIUT    FKOM    VUVn    ui;sKI!VATI()NS. 


283 


ami  /'"  arc  involved  in  tlioni,  tlicv  will  cnal)li'  us  tu  dctcrniinc  tlit'sc 
cuitiil*'  distances. 
Lit  us  now  put 

cos ,j'  sin  (A'  —  k)    =A, 

(•(.s,J"i.in(A"'— r):^.  C, 

all''  the  iM'cccding  oijuations  give 


cos  ,j"  sin  (■/"-;.)  =n, 


COS -J  j-in  i  A 


(6) 


,,,,'  sfc  ,j'  —  /;«"//'  sec  ,J"  -  -  nli  sin  {k—Q)  —  Ji'  sin  (/>.  —  ©') 

+  /."A"'sin(/i. - 
/>//,-'  scc,i'-  f//'  8cc,J"=-  n'Ji'  sin  (/'"-  ©')  -  /^  sin  (/.'"-  ©") 


O"), 

<7) 


+  //"7r.sin(;."'-o"'). 

If  we  assume  for  n  and  n"  tlioir  values  in  the  ease  of  the  orliit  of 
tlic  earth,  which  is  {'(luivalcnt  to  ncglcctin;^  terms  of  the  sccctnd  order 
in  llic  c(|iiaiions  (-0).,,  the  .sccon«l  mcnduT  of  tlw  first  of  these  e(jua- 
tiiiii<  reduces  rigorously  to  zero ;  and  in  the;  same  manner  it  esm  he 
slutwn  that  when  sindlar  terms  of  the  second  order  in  the  corre- 
t.l)nnding  expressions  for  n'  and  n"  are  neglected,  the  second  mendier 
of  the  last  cijuation  reduces  to  zero.  Jlence  the  sec<tnd  mend>er  of 
each  of  these  etiuatious  will  generally  differ  from  zero  hy  a  t|uantity 
wliicli  is  of  at  least  the  second  order  with  respect  to  the  intervals  of 
time  hetween  the  ohservations.  The  eoeflicients  of  (i'  and  />"  are  of 
the  lirst  order,  and  it  is  easily  seen  that  if  we  eliminate  //'  froui 
those  eijuations,  the  resulting  e(|uation  for  //  is  such  that  an  error  of 
tile  secdud  ()rder  in  the  values  of  /i  and  ii"  may  produce  an  error  of 
tlu'  order  zero  in  the  result  for  »/,  so  that  it  will  not  he  even  an 
appr  >.\iniation  to  the  eorreet  value;  and  the  same  is  true  in  the  case 
of  //'.  It  in  necessary,  therefore,  to  retain  terms  of  the  second  order 
in  the  fii-st  assumed  values  for  /),  n',  n",  and  n'" ;  and,  .since  the 
terms  of  the  second  order  involve  /•'  and  /■",  we  thus  introduce  two 
a<Mitional  uid<nown  (piantities.  Ilep-ce  two  a<lditional  e<piations  in- 
volving /•',  /■",  ft',  It"  and  (piantities  derived  ii\»m  ohservation,  must 
he  obtained,  so  that  hy  elinuuation  the  values  of  the  quantities  sought 
may  he  found. 
From  e(|uation  (04)^  we  have 


//  see  yS'  =:  It'  cos  4'  ±  ]/  r"  -  Ji'^  sin'  +', 


(8) 


wiiieh  is  one  of  tiie  equations  required;  and  similarly  we  find,  for 
the  other  equation, 


p"  sec  ,r  =.  Ii"  cos  4"  ±  V  r"'-  K*  siu»  +". 


(9) 


y; 


284  TIIKORETICAL   A8T10N0MY. 

Intrwlucing  these  values  into  the  equations  (7),  and  putting 


wo  get 


x*  =±l/r"-/;"Mn»4?, 


(10) 


Ax'  —  £h'V'  =  nR  sin  (X  —  Q)  —  If  sin  {X  —  ©') 

-f  n"U"  m\  {X  -  O" )  -.  AW  cos  V  +  n"JS/?"  cos  4", 
Dn'x'  -  Cx"  =  n'li'  sin  (A'"  -  ©')  —  li"  m\  O-'"  -  G") 

+  H"'ii"'  sin  (a'"  -  0'")  -  n'DW  cos  V  +  C'ii"  cos  4". 


Let  us  now  put 


l-h' 


D 

C 


h", 


or 


h' 


cos  fi"  am  (X"~X) 
cos/S'smU'  — A)' 


A"  _  co8;?'sin(;r-/i') 

^  ""co»/3f"8iuCA"'-/l"y 


,.,  .  .   7?' sin  (;.  -  ©') 

if  cos  4  H . =^  o , 


JJ"C08  4"- 


^1 

/rsinCr^-O") 

c 


a 


(11) 


i?sin(A  — 0) 


=  d', 


C 

JR'"  Bin  (X"'—0"') 
C 


=  -  d", 


and  we  have 


a/  =  h',i"x"  4-  ntT  -  a'  -+  n"c', 
«"  =  h"n'x'  +  n"'d"  -  a,"  +  n'c". 


(12) 


These  equations  will  serve  to  determine  x'  and  .r",  and  hence  /'  auJ 
r",  as  soon  as  the  values  of  n,  »',  n",  and  n'"  are  known. 

96.  In  order  to  include  terms  of  the  second  order  in  the  valuos  of 
n  and  n",  we  have,  from  the  equations  (26),, 

and,  putting 


these  give 


^  —  n'" 


Q'  =  in  +  n"-l)r^, 


(13) 


OKuiT  fro::  four  oiwervations. 


Lot  us  now  put 


r^'=.k(e"-t'), 


285 

(14) 

(15) 


011(1,  mnkiiig  the  ncccssury  changes  in  the  notation  in  (Hjuntions  (2G)„ 
we  obtain 


«"'=r'(n-i 

'o   ' 


r"'(T;4-T)  T"'(T""-|-r"'T-T»)    f/r" 


«"•» 


-i 


kr"* 


(16) 


"    ~<\^'^"*         r'"      "  "•"  •  '  /fcr''*  '"(it'"!' 

From  these  we  get,  inohiding  terms  of  the  second  order, 


and  hence,  if  we  put 


n 


P"  —  :■_ 


n 


Q"  =  (n'  +  7i"'-l)r">, 


(17) 


wo  shall  have,  since  To'  -=  r  +  r"', 

■P"  =  -.,,  ( 1  ~  i  — /7i — 1» 

When  the  intervals  are  equal,  we  have 


(18) 


P'— — 


P"—  _ 


and  these  expressions  may  be  used,  in  the  case  of  an  unknown  orbit, 
for  the  first  approximation  to  the  values  of  these  quantities. 
The  equations  (13)  and  (17)  give 


n  =  n"P'', 

n"'  =  n'P"; 
and,  introducing  these  values,  the  equations  (12)  become 


(19) 


^^ 


d^i 


286 


TIIF.OHKTKAI-    AHTKOXOMY. 


*  =   1  +  />'  (  1  +    '^'  )  '  /''•'•"  +  ^"'''  +  ^')  -  «'. 

*"  =  1  +  P"  (  1  +  ^C  )  C/'"-^'  +  i^"''"  +  c")  -  a". 
Let  us  now  put 


P'd'-\-e' 


1  +  / 

f " 
n 


J'  — /  » 


i+1 


ill  — J  f 


and  \vc  shall  have 


Wc  have,  further,  from  equations  (10), 


(20) 


(21) 


(22) 


(28) 


If  wo  substitute  these  values  of  ?•'•'  and  r"^  in  equatims  (22),  the  two 
resultiuff  equations  will  contain  only  two  unknown  (juautiiies  .<•'  ;iiul 
x",  when  P',  P",  Q',  and  Q"  are  known,  and  hence  iliey  Mill  l)e 
suiKcient  to  solve  the  problem.  IJut  if  we  elVe<>t  the  eliminatinn  <if 
either  of  the  miknown  quantities  directly,  the  re.sultinjr  (Hiuation 
becomes  of  a  hijjjh  order.  It  is  necessary,  therefore,  in  the  numerical 
ap|)lication,  to  solve  the  ecpiations  (22)  by  successive  trials,  which 
may  be  readilv  ertccted. 

If  z'  represents  the  anjjlc  at  the  planet  between  the  sun  and  the 
earth  at  the  time  of  the  second  observation,  and  z"  th.e  same  angle  at 
the  time  of  the  third  observation,  we  shall  have 


,      R'  sin  4' 


sms 
/?"sinV' 


(24) 


smz 


// 


Substituting  these  values  of  r'  and  r"  in  equations  (10),  we  get 


and  hence 


a/  =  r'  "OS  z', 
»"  =  r"cos2", 


(25) 


OR  HIT    FUOM    FOLK  0HSEUVATI0X8. 


287 


tan?  = 


h'  sill  V 


tun  z"  =^  — ^,^- 


(26) 


l)v  moans  of  wliioh  wo  nmy  find  ;'  and  ;"  as  soon  as  .r'  un<l  .v"  shall 
liiivc  boon  (lotcrnunod ;  a:^!  then  /•'  ami  /•"  an;  obtained  I'roni  (.'4)  or 
(2-")).  The  last  cqnations  show  that  when  x'  is  nt'ji;ativo,  rJ  mint  bo 
grcator  than  90°,  and  hoiico  that  in  this  case  r'  is  loss  than  W. 

In  th(!  nuinorioal  applii-ation  of  o(|uations  (22),  for  a  first  apjtroxi- 
niatiun  to  tho  vabics  of  ,/•'  and  .r",  since  (/  and  (/'  are  quantities  of 
tho  second  order  with  resi)eet  to  r  or  r'",  wo  may  generally  put 


and  we  have 


or,  hy  elimination, 


<2'  =  o. 


<?"-0; 


1  -ff 
»_'-o"+Ao'-/"«'- 


- "       l-f'f" 

With  the  approximate!  values  of  .r'  and  .r"  derived  from  these  oqua- 
tiitns,  we  compute  first  /•'  and  /•"  from  tho  orpiations  (2(5)  and  (21), 
ami  then  new  values  of  .r'  and  .r"  from  (22),  tho  o])eration  iM-ing 
repeated  until  tho  true  values  are  obtained.  To  facilitate  those  ap- 
proximations, the  equations  (22)  give 


X    = 


/('+^:)  ^' 


(27^ 


7fr+?;:)  ^"' 


liCt  an  approximate  value  of  x'  be  designated  by  x^,  and  let  the 
value  of  x"  derived  from  this  by  moans  of  the  first  of  equations  (27) 
be  designated  by  .r„".  With  the  val;><3  of  Xq"  for  x"  wo  derive  a 
new  value  of  x'  from  the  second  of  those  equations,  which  we  denote 
by  .1'/.  Then,  recomputing  x"  and  x',  we  obtain  a  third  approximate 
value  of  the  latter  quantity,  which  may  be  designated  by  x./ ;  aid, 
if  wo  put 


•''1  •''0   —  ^0' 


a;,'  =  a/, 


288 


TlIKOIlCTICAIi   AHTHOXOMl , 


W(?  sliall  Imvo,  norordiiij;  to  tlic  tHjimtion  (67),,  the  ncccR'>n:y  clianj^cs 
being  made  in  the  nutation, 


«'  =  x/- 


(28) 


The  vahie  of  x'  thns  obtained  will  give,  by  means  of  the  first  of 
e(juations  (27),  a  new  vahie  of  a*",  and  the  .snbstitution  of  this  in  the 
last  of  these  eijuations  will  show  whether  the  correct  result  has  Im-cu 
foun<l.  If  a  repetition  of  the  cidciilation  Ihj  found  ne«!essary,  the 
three  values  of  x'  which  appntximate  nearest  to  the  true  value  will, 
by  meai.s  of  (28),  give  the  correct  result.  In  the  same  manner,  if 
we  assume  for  x"  the  value  derived  by  putting  (/  ~-0  and  (^"--0, 
and  comi)Ute  x',  three  successive  approximate  results  for  x"  will 
enable  us  to  interpolate  the  corre(!t  value. 

When  the  elements  of  the  orbit  are  already  approximately  known, 
the  first  assumed  value  of  x'  should  bo  derived  from 


«'  =  >//'— /e'^siu'V 
instead  of  by  putting  Q'  and  Q"  equal  to  zero. 

97.  It  should  be  observed  that  when  ^'  =^  ^  or  X'"  =  ?-",  the  equa- 
tions (22)  arc  inapplicable,  but  that  the  original  equations  (7)  give, 
in  this  case,  either  p"  or  //  directly  in  terms  of  n  and  n"  or  of  it,' 
and  n'"  and  the  data  furnished  by  observation.  If  we  divide  the 
first  of  equations  (22)  by  /«.',  we  have 


h! 


':={^+m-^^H 


The  equations  (21)  give 


I 
h' 

and  from  (11)  we  get 

a! 

e  _ 
K~ 

K~ 
Then,  if  we  put 


1 

1  +P" 


''    1  +  P'  ' 


/?  cos  4'   ,  K  sin  (X  —  0') 
H 


h' 


B 

iJ"sin(>l- 


O") 


i?8iu(/l  —  0) 


c: 


^/I'  +  F 


(29) 


ORBIT   FROM    FOITR  OIWERVATION8. 


289 


its  Viiluo  may  Im-  fiumd   f'nuu  the  rcsiiltH  foi*     ,  and    .,  durivcd  l)y 
meann  of  tht'j't'  cHjiiatiniiis,  and  \\v.  shall  have 

\\\\v\\  X'  ~-  X,  wc  have  h'  =  oc,  and  this  formula  becomes 

the  value  of   -,  l>eiuj;  fjiven   hy  the  first  of  equations  (20)      This 

C(|Uiition  and  the  second  of  e(|iiations  (2U)  are  sufTieient  to  determine 
3-'  and  .!•"  in  tin-  s|H>cial  ejisc'  under  eonsidi'ration. 

The  see(»nd  of  ecjuatious  (22)  may  be  treate<l  in  precisely  the  same 
niiiuner,  so  that  when  k'"  =-  X",  it  becomes 

and  this  must  be  solved  in  connection  with  the  first  of  these  equations 
in  order  to  find  x'  and  x". 

98.  As  soon  as  the  numerical  values  of  x'  and  .r"  have  Iwen 
derived,  those  of  /•'  and  r"  may  be  found  by  means  of  the  equations 
(26)  and  (24).     Then,  according  to  (41)^,  we  have 


p'  =  — - .    , cos  /y, 

.sm  2 

,,      /2"8inf2"+V')_^, 
P  = 7.v::zfi cos  p  . 


(31) 


sinz 


Tlic  heliocentric  places  are  then  found  from  p'  and  ft"  by  means  of 
tile  niuations  ("l)^,  and  the  values  of  r'  and  /•"  thus  obtained  should 
atrrcr  with  those  already  derive<l.  From  these  places  we  compute 
the  i)osition  of  the  plane  of  the  orbit,  and  th(;nce  the  arguments  of 
the  latitude  for  the  times  t'  and  t". 

The  values  of  /•',  /•",  u',  u",  n,  n",  w',  and  n'"  euable  us  to  deter- 
miuc  r,  >•'",  h,  and  it'".     Thus,  we  have 

[r'r"]  =  rV"  sin  («"-«), 
and,  from  the  equations  (1)  and  (3)j, 

19 


290 


THKOllKTK  A 1.   AHTIIONOMY. 


n 


/ 


M    =-^,-t>-V'], 


n 


Therefore, 


[rr"j    -^llr'r"], 
[f"n  -■     'i  Ir'r"], 


n 


r  sin  ( u'  —  u)      =  —  r"  sin  («"  —  «'), 


n 


rsni 


(„"  _  „)     ^l  /  Bin  („"  _  u'), 
n 


n 


(32) 


r'"  sin  (u'"-  u")  =  4;  /  sin  (u"  -  «'), 
n 

r'"  sin  (u'"  -  u')  =--  1,  »•"  sin  (u"  -  «')• 

From  tl».»  fir.st  un'>l  .second  of  these  equations,  by  addition  and  ^^ub- 
trac'tion,  we  get 


rsin 


((h'  -  «)  +  i  (n"  -  «'))  =  '^+-?^'  sin  i  (n"  -  «'), 
r  cos  ((it'  —  tO  +  \  (u"  —  «'))  =  -=^^'  cos  J  (n"  -  «'), 


('33) 


from  whieli  we  may  find  r,  u'  —  «,  and  n  ^=  u'  —  [u'  —  m). 

In  a  simihir  manner,  from  the  third  and  fourth  of  equations  (32), 
we  obtain 


r"'  sin  ((«'"  —  It")  +  {  (u"  -  u'))  =  --+,—  sin  A  («"  -  it'), 

IV 

r'"  co8((n"'  -  It")  +  {  («"  -  «'))  =  ^,7—  cos^  (it"  -  tt'), 


(34) 


/  ,,"  ,,"' 


from  whi(!h  to  find  r'"  and  it'". 

When  the  approximate  values  of  v',  r',  ?•",  )•'",  and  tt,  «',  tt", 
have  been  found,  by  means  of  the  preceding  equations,  from  the 
assumed  vahies  of  P',  P",  Q',  and  Q",  the  second  approximation  to 
the  elements  may  be  commenced.  But,  in  the  ease  of  an  unknown 
orbit,  it  will  be  exiKnlient  to  derive,  first,  approximate  va.ues  of  /' 
and  r",  using 


P'  — _- 


P"  = 


T^ 


and  then  recomijute  P'  and  P"  by  means  of  the  equations  (14)  and 


OniUT   FIIOM   FOrn  OUHFUVATroNS. 


2fll 


(|H),  licfori'  fitxiiii);  ii'  ami  ii".     The  Utiiis  of  tln'  wcond  (»nl«'r  will 
tliii.H  l)c  coiiiplctcly  taken  into  account  in  the  iii'^t  approxiniatiun. 

W.  If  the  tiiiH'.M  of  ohscrvation  have  not  U'on  «'orrc<'t«M|  for  the 
time  of  aiH'i-ration,  as  in  the  case  of  an  urhit  wholly  unknown,  this 
currection  may  he  applietl  iH'fore  the  secon<l  approximation  to  tiie 
elements  is  eiteetcd,  or  at  least  hefore  the  fnial  approximation  is  eoni- 
iiieiieed.  For  this  purpose,  the  distances  of  tin-  hody  trom  the  earth 
fl»r  the  four  oi)servali<ius  must  he  determined ;  and,  since  the  curtate 
distances  />'  and  (»"  arc  alrcatly  jjiven,  there  remain  only  //  and  //"  to 
lie  tiiund.  If  wo  eliminate  />'  from  the  first  two  of  (H^'iations  (.'I),  tlu* 
result  is 


'-^f 


„n"m\(X"—k') 


('.]r» 


n  sin  ( A  —  i) 

nit 8in(A'  — G)  — /e' sin  i^'—Q,   |- /t"A'"Mn  fA'  — 0")_ 
-r '       -    '  ^j^i^jii  (^k'  ~  k)  '  ' 

uhI,  hy  eliminating  ,o"  from  the  last  two  of  these  ofpiations,  we  also 
u)>tuin 


(36) 


,n' Bin  (/';;— _A') 
"  n"'m\(i"'~X") 
n' Ji' »\u  {k"  -  Q')  ~  Ji"  urn  C/l"— CT)  +  n'"  Ii'"  ain  (k"  -  Q^'^ 

n"'miU"'-X") 


hy  means  of  which  p  and  //"  may  he  found.     The  combination  of 
the  lirsl  and  second  of  equations  (.'])  gives 


P~^  cos  (A'— A) 


n"p' 
n 


cos(r— ;) 


(37) 


+ 


nl{  cos  (X  —  Q)~Rcos(i  —  ©')  -f  n"  R"  cos  {k  —  ©") 


n 


and  from  the  third  and  fourth  we  get 


V;^co8(r'-r)-'|-^:'co3(r 


n 


(38) 


+ 


n'  Ii' cos  (A'"—  0')  —R"  cos (X"'~  Q" )  -\-n"' R'"  coii(X"'—0"') 


n" 


Further,  instead  of  these,  any  of  the  varioiis  formuhe  which  have 
been  given  for  finding  the  ratio  of  two  eurta^^  distances,  may  be 
employed;  but,  if  the  latitudes  /9,  ^9',  &e.  are  very  small,  the  values 
of  f)  and  p'"  which  depend  on  the  differences  of  the  observetl  longi- 
tudes of  the  body  must  be  preferred. 


292 


TIIKOUKTKAr-    AHTRONOMY. 


The  values  of  //  and  p'"  may  also  lie  <l«'i'iv<'(l  hy  coinpiitin^  tin- 
heliocentric  places  of  tlu;  ImhIv  for  the  tinu'fj  /  and  t'"  hy  means  of 
till!  e<|uations  (H2)i,  and  (hen  finding  the  getM-entric  places,  or  (hose 
whicli  helonj;  to  the  points  to  which  the  ohservation.s  hav(!  hccn 
ntduct'd,  hy  means  of  (!)())i,  writinj^  (>  in  place  of  J  cos  ^9.  'i'iijs 
process  all'ords  a  verificali<»n  of  the  inimerical  caI(;ulation,  nainelv, 
the  Vidnes  of  X  and  X'"  thus  found  .should  a^rei;  with  those  furnished 
l)y  observation,  and  the  a^re(!ment  of  the  computed  latitudes  ^9  and 
[•i'"  with  those  ohserved,  in  <us(!  the  latter  are  ^^iven,  will  show  how 
iKnirly  the  position  of  the  phine  of  the  orbit  as  derived  from  tli(! 
second  and  third  ol)servations  represents  the  extreme  latitudes.  If 
it  were  not  desirable  to  compute  /  and  /'"  in  order  to  check  tlic 
calculation,  even  when  ,9  an<l  {i'"  are  {^iven  by  observation,  wc  might 
derive  /;  and  //"  from  the  e(iuation» 


p    =r  Bin  H  fin  /  cot/?, 
/r,"'=ur"'8in(("'sin/cot,J" 


(39) 


wlien  the  Intitudes  are  not  very  small. 

In  the  linal  approximation  to  the  element.s,  and  especially  when 
the  position  of  the  |»Iane  oi"  the  orbit  <-annot  be  obtaiiKHl  with  the 
recpiired  |)recision  from  the  ,seeon<l  and  third  observations,  it  will  he 
ailvantai^eous,  provid(>d  ti  .  the  data  furnish  th(.'  extreme  latitudes 
fi  and  ^9'",  t<»  compute  />  and  //"  as  .soon  as  (>'  and  //'  have  iiicii 
found,  and  then  find  /,  /"",  h,  and  />'"  directly  front  tliese  by  means 
of  the  formula'  (71),.  Tlu;  values  of  Q,  and  /  may  thus  hi\  obtained 
from  the  extreme  places,  or,  tlu;  heliocentri<!  places  for  the;  tiniest' 
and  /'"  beinjf  also  computed  directly  from  //  and  //',  f'ntm  those 
which  arc  best  suited  to  this  purpo.se.  Hut,  since  ti\e  data  will  he 
nior(!  than  suiricient  for  the  solution  of  the  problem,  when  the  exlrenio 
latitudes  are  use<l,  if  we  coniput(!  the  heliocentric  latitudes  h'  and  //" 
from  the  eipiatious 

tan  //     -  tan  /  sin  iH  —  SJ), 
tan  b"  =  -  tan  i  sin  (l"  —ft), 

they  will  not  aii;re(>  exactly  with  the  result.s  obtained  directly  from  /»' 
an<l  ,'/',  unless  the  four  observations  are  completely  satisfied  by  the 
elements  obtained.  The  values  of"  r'  and  /•",  however,  (!omputetl 
directly  from  (>'  and  jt"  by  niean.s  of  (Tl).,,  nuist  agree  with  those 
derived  from  .r'  and  .r". 

The  corrections  to  be  applied  to  the  times  of  observation  on  account 


OKIMT    FROM    FOUR   OBSKRVATIONS. 


293 


of  :il«'rrntion  may  now  l)o  foiiiid.     Tims,  if  /„,  /„',  /„",  and  /,/"  are 
the  uiioorrecttstl  thncs  of  observation,  the  (rorre<!t<.'<l  values  will  be 


t    =/.  ^    r/*see,?, 


(40) 


e"  =  t^" 


wherein  \o\rC--  '7.7Wh2-\,  luul  from  these;  \v<!  (U'rive  tiie  cftrreeted 
values  of  r,  r',  7",  t'",  and  r,,'. 

KK).  To  find  the  values  of  /",  /'",  Q',  and  (/',  which  will  bo 
vKwt  when  /•,  /•',  r",  r'",  and  »,  »',  a",  n'"  are  aeenriitcly  known,  we 
have,  aeeordin;;  to  the  ('(juations  (17)^  and  {J'A)„  since  </       ^Q, 


1>'.. 


Q  -  i  L  » 


(41) 


«7'  ■  rr"  cos  i  ( 11"  —  'u!)  eos  ^  ( 11"  —  v)  cos  A  («'  ■  -  « )' 


In  a  similar  manner,  if  we  desij;nate  by  m'"  i\u>  ratio  (»f  the  sector 
formed  i)y  the  radii-vectores  r"  and  r'"  to  tin;  trianj^le  formed  by 
the  same  radii-vectores  and  the  chord  joining  their  extremities,  we 
tlnd 


P"  ^_  — 

■*         ' lit 


Q" 


TT 


J>* 


(42; 


*  .W"    rV"  C08  \  ill!"  -  «")  (!08  ^  («'"  -  n')  cos  i  («"  -  n') 


The  formuhc  for  finding  the  value  of  s'"  arc  obtjiined  from  those  for 

«  by  writing  x"'y  X"'y  ^'"">  *^'*'-  '"  p'^^*'^'  "*  Jf'  ?'»  ^'>  ^'^'"'^  '^'•''  ""^ing 
}•',  /•'",  \i!"  —  »,"  instead  of  /•',  /•",  and  \i"  —  n\  respeejtively. 

By  means  of  the  results  obtained  from  th(!  first  approximation  to 
tile  values  of  /",  P" ^  Q',  and  (j>",  we  may,  from  ('(piations  (11)  and 
( 12),  deriv(!  new  and  more  nearly  accuriite  values  <»f  these;  (|uantities, 
and,  by  ri']K!ating  the  culeidation,  the  approxiniations  to  the  exact 
values  mu  •  be  earrieil  to  any  extent  whic^h  niay  be  desirable.  When 
tlu-e(!  approximate  values  of  P'  and  (/,  and  of  P"  and  C/',  have 
been  derived,  the  next  approximation  will  l)e  fiu'ilitated  by  the  use 
of  the  formula'  (H2)^,  as  already  explained. 

\Vh<.!n  the  values  of  7^',  P",  Q',  and  Q"  have  been  derive(l  with 
sullicieiit  aeeuracy,  we  proceed  from  these  to  find  the  elements  of  the 
orbit.  After  S2,  /,  r,  /•',  r",  r'",  «,  u',  n",  and  n'"  have  been  found, 
l.ie  remaining  elements  may  be  derived  from  any  two  radii-vectores 


294 


THEORETICAL   ASTRONOMY. 


and  the  corrospoiuling  arf^mnents  of  the  hititude.  It  will  be  most 
aeciirate,  however,  to  derive  the  olcnients  from  r,  r'",  u,  and  it'". 
If  the  vahies  of  F',  P",  (}',  and  Q"  liave  been  obtained  with  groat 
aeeuraey,  tlie  res-.ilts  lierived  from  any  two  places  will  agree  with 
tho!<(!  obtained  from  the  extreme  places. 
In  the  first  j)lace,  from 


sin^j  cos  Gj  =  sin  ^  («" 


sin  Y^  sin  6rj 
cos  r^ 


cos^fw" 
cos  A  (n" 


u), 

u)  cos  2/o, 
-  u)  sin  2xo, 


(43) 


we  find  Yq  and  Gq.    Then  we  have 


r,=^k(f 


t), 


W'o  =  ' 


(,.  + /")S  cosVo 


»»» 


COS)'/ 


(44) 


'?o 


Jo» 


0  4"./o  "1""  ^0 

from  which,  by  means  of  Tables  XIII.  and  XIV.,  to  find  s^^  and  x^. 

We  have,  further, 

"'"sin  («'"—«)' 


P 


>oZ^ 


')■• 


and  the  agreement  of  the  value  of  p  thus  found  with  the  separate 
results  for  the  same  quantity  obtained  from  the  combination  of  any 
two  of  the  four  places,  will  show  the  extent  to  which  the  npi)roxinia- 
tion  to  P'y  P",  Q',  and  Q"  has  been  carried.  The  elements  are  now 
to  be  computed  from  the  extreme  places  jn-ecisely  as  explained  in  the 
preceding  chapter,  using  r'"  in  the  ])lace  of  r"  in  the  formula)  there 
given  and  introducing  the  necessary  modifications  in  the  notation, 
which  have  been  already  suggested  and  which  will  be  indicated  at 
once. 

101.  Example. — For  the  purpose  of  illustrating  the  application 
of  the  fornnihe  for  the  calculation  of  an  orbit  from  four  observations, 
let  us  take  the  following  normal  places  of  Eurynome  @  derived  by 
comparing  a  series  of  observations  with  an  ephemeris  computed  from 
approximate  elements. 


Irreenwich  M.  T. 

1863  Sept.  20.0 
Dec.  9.0 

1864  Feb.   2.0 
April  30.0 


14°  30'  35".6 

9  54  17  .0 

28  41  34  .1 

74  29  58  .9 


+  9°23'49".7, 
2  53  41  .8, 
9  6  2  .8, 

+  19  35  41  .5. 


NUMERICAL   EXAMPLE. 


295 


Those  norniiils  give  the  geocentric;  phiee-s  of  the  planet  referred  to  the 
mean  equinox  ami  equator  of  18(!4.(),  and  free  from  aberration.  l'\)r 
the  mean  obliquity  of  the  eeliptic  of  18()4.0,  the  American  Nautical 
Almanac  gives 

e  =  23°  27'  24".49, 

and,  by  means  of  this,  converting  the  okserved  right  ascensions  and 
declinations,  as  given  by  the  normal  places,  into  longitudes  and  lati- 
tudes, we  get 


(irccinvich  M.  T. 

1803  Sept.  20.0 
Dec.   9.0 

1804  Fob.   2.0 
April  30.0 


?. 
16°  59'  9".42 
10  14  17  .57 
29  53  21  .99 
75  23  46  .90 


+  2°  56'  44".58, 

—  1    15  48  .82, 
2    29  57  .38, 

—  3      4  44  .49. 


Those  places  are  referred  to  the  ecliptic  and  mean  equinox  of  1864.0, 
and,  for  the  same  dates,  the  geocentric  latitudes  of  the  sun  referred 
also  to  the  eeliptic  of  1864.0  are 


H-  0".60, 


+0".53, 


+  0".36, 


+  0".19. 


For  the  reduction  of  the  geocentric  latitudes  of  the  planet  to  the 
jwiiit  in  which  a  perpendicular  let  fall  from  the  centre  of  the  earth 
to  the  plane  of  the  ecliptic  cuts  that  plane,  the  equation  {^^)^  gives  the 
corrections  —  0".57,  -0".38,  —  0'M8,  and  —  0".O7  to  bt;  a|)plied  to 
these  latitudes  respectively,  the  logarithms  of  the  approximate  dis- 
tances of  the  planet  from  the  earth  being 


0.02618, 

0.13355, 

0.29033, 

0.44990. 

Thus  we  obtain 

t   =     0.0, 
/'  =    80.0, 
r  =.135.0, 
r  =  223.0, 

;    =16°  59'    9' 
A'   =10    14  17 
k"  =  29    5.3  21 
A'"  =  75   23  46 

.42, 

.57, 
.99, 
.90, 

<5    = 

=  -|-2°56'44".0], 
-  —  1    15  49  .20, 
=  —  2    29  57  .5t), 
=  —  3      4  44  M ; 

and,  for  the  same  times,  the  true  places  of  the  sun  referred  to  the 
mean  equinox  of  1864.0  are 


O  =177°  0'58".6, 
0'  =  256  58  35  .9, 
O"  =312  57  49  .8, 
0'"  =   40   21  26  .8, 


log  ft  =0.0015899. 
log  R'  =  9.9932638, 
log  li"  =  9.9937748, 
log  R"'  =  0.0035149, 


-' 


y 


206 


THEORETICAIi   ASTRONOMY. 


From  the  ecjuations 

_  tan/5' 

sin(/l'— ©')' 
_  tan  (5" 
sin  (/. 

we  obtain 


tanitf  ■■ 
tan  w" : 


^      ,        tan  (/'-©') 

tan  4  = ; , 

cosij; 

tan  (/"  -  O") 


TTJ 


O") 


tan  4"  = 


v"08  tV 


V  --^  113°  15'  20'M0, 
V'=   76    56  17  .75, 


log  (/r  cos  4')  =9.58nfi777„, 
log(/ir  siuV)  =9.9564624, 
log  (/r  cos  V  j  =  9.a47«.S48, 
log(ir8in+")  =  9.9823904. 


The  quadrant  in  wliich  -vl/'  mu.st  be  taken,  is  indicated  by  the  condi- 
tion that  eot«\|/'  and  co.s(/' — G')  must  have  the  same  .sign.  The 
same  condition  exist.s  in  the  ca.se  of  ■»//".     Then,  the  formuhe 


A  — •  cos  /5'  sin  (X'  —  A), 
C=  cos /i"sin(r '—/"), 

B 

A~"'  C 

R  sin  (>l  —  ©') 


h', 
a!  =  Ji'  cos  4'  + 


^=rcos;5"sin(A"  — A), 
Z)  =  cosrj'sin(A"'— A'), 


a"  =  i?"cosV'- 


d' 


c'  =/t'iJ"co8  4"  + 

c"  =  h"R  co^A,'  — 

R  sin  (A— 0) 
A  ' 


R"^m{r—Q") 
C 
R'mxU  —  Q") 
~~A  ' 

/e's  i(r'-o') 


d": 


i?"'sin  r'— ©'") 


give  the  following  results: — 

log^=:^9.0699254„, 
log  B  =  9.3484939, 
log /i'  =  0.2785685,, 
loga'  =  0.8834880„, 
logr'  =0.9012910„, 
log  d'  =  0.4650841, 


log  C  =  9.8528803, 
log  Z>  =  9.9577271, 
log  r  =  0.1048468, 
logrt"  =  9.9752915,,, 
logo"  =9.7267348.,, 
logcZ"  =  9.9096469„. 


We  are  now  pi'cpared  to  make  the  first  hypotheses  in  regard  to  the 
values  of  P',  Q',  P",  and  (/'.  If  the  elements  were  entirely  un- 
known, it  would  be  necessary,  in  the  first  instance,  to  assume  for  these 
quantities  the  values  given  by  the  expressions 


NUMEPiCAL   EXAMPLB. 


297 


P'=  — 


P"  _  _!_ 


tlipn  approximaio  values  of  /•'  and  r"  are  readily  obtained  by  means 
of  the  c<iuations  (27),  (2G),  and  (24)  or  (25).  The  first  assumed 
value  of  x'  to  be  used  in  the  seeond  mt-ml)('r  of  the  fii*st  of  e([uati(ms 
(27),  is  obtaineil  from  the  expression  whieh  results  from  (22)  l>y 
putting  Q'  =  0  and  (^"  —  0,  namely, 


«'  = 


<+/V-A"- 


1  -rr 


after  which  the  values  of  x'  and  x"  will  be  obtiiined  by  trial  from 
(27).  It  should  be  remarked,  further,  that  in  the  first  determination 
of  an  orbit  entirely  unknown,  the  intervals  of  time  between  the  ob- 
ficrvations  will  generally  be  small,  and  hence  the  value  of  x'  derived 
from  the  assumption  of  (^' =- 0  and  (^""--Owill  be  sufficiently  ap- 
proximate to  facilitate  the  solution  of  equations  (27). 

As  soon  as  the  approximate  values  of  r'  and  r"  have  thus  been 
found,  those  of  P'  and  P"  must  be  recomputed  from  the  expressions 

With  the  results  thus  derived  for  P'  and  P",  and  with  the  values  of 
§'  and  Q"  already  obtained,  the  first  approximation  to  the  elements 
must  be  completed. 

When  the  elements  are  already  approximately  known,  tiie  first 
assumed  values  of  P',  P",  Q',  and  Q"  should  be  computed  by  means 
of  tlicse  elements.     Thus,  from 


n  = 


rV'sin(v"  — v') 


rr"  8in(v"- 


n" 


rr'  sin  (v'  —  v) 


n' 


r"r"'^m{v"'~v") 
m\{v"'—v'y 


rr"  ^h\{v"  - 
rV'sinrv" 


-v') 


rr 


^    ~i'r"''sm{v"'-v')' 


we  find  n,  n',  n",  and  n'".     The  approximate  elements  of  Eunjnome 

give 

V    =322°  55'    9".3,  logr    =0..308327, 

v'  =353    19  26  .3,  log/   =0.294225^ 

v"  =    14    45    8.5,  logr"  =0.296088, 

v"'=   47    23  32.8,  log/"  =  0.317278, 


\il 


298  TIIEOKETICAL   ASTRONOMY. 

and  heiiec  we  obtain 


Tlien,  from 


we  get 


logn  ^  9.rM,']052, 
log  h'  ~  9.825408, 


•*    —  ~iri 
n 

n'" 

n 


log  P'  =^  9.84G216, 
log  P":^  9.807763, 


log  n"  ---^-  9.806836, 
logn'"  =^9.633171. 

(2'  =(„  +  »" -!),.'», 
^'  =  (n'+  n'"-  1)  r"», 

log(^  ^9.840771, 
log  (/'=.  9.882480. 


The  values  of  these  quantities  may  also  be  computed  by  means  of  tiie 
equations  (41)  and  (42). 


Next,  from 


we  find 


P'd'  +  c! 

I  +'P'"' 
W'-l-  c" 

l  +  i 


^•d  —   1  j^'pn 


^  ~\+  P'' 

h" 


[ill     '  J  11       !>"• 


log..'  =.0.041 344„, 
log  r„"  ---=  9.807665„, 


log/'  =  0.047658,, 
log/"  =  9.889385. 


Then  we  have 


■r'  +  q'  r; 

^(1+^;)  •^"^' 

x"  +  a" 


a;  = 


r(i+:q  ^ 


0 


tan  2'  = 


/  = 


i?'  sin  V 
x'      ' 
ii'  sin  +' 


tan  2 '  = ,y-^-, 

X 

„      i2"sin4" 


sin/         cos  2'  sin  2"  cos  2" 

from  which  to  find  r'  and  r".     In  the  first  place,  from 

x'  ^-.  l//»— i2"sinH', 
we  obtain  the  approximate  value 

log  a;' =  0.242737. 
Then  the  first  of  the  preceding  equations  gives 

loga;"  =  0.237687.       - 


NUMERICAL   EXAMPLE.  299 

From  thi.s  we  get 

2"  =  29°  3'  11".7,  log  (•"  ^  0.296092 ; 

ami  then  the  equation  for  x'  gives 

log  x'^  0.2427(58. 
Heuce  we  have 

^  =  27°  20'  59".6,  log  r'  =  0.204249  ; 

and,  rcpouting  the  operation,  using  these  rcsult.s  for  x'  and  r',  we  get 

log  .1;"=-- 0.237678,  log  .c' =  0.242757. 

TliP  correct  value  of  log.)'  may  now  be  found  hv  inean.s  of  equation 
(28).    Thus,  in  units  of  the  sixth  decimal  place,  we  have 


fl„  =  242768  — 242737=^  +  31, 


a,'  =  2427o7  -  242768  ^  —  11, 


and  for  the  correction  to  he  applied  to  the  last  value  of  log  .r',  in 
units  of  the  sixth  decinuil  place, 


A  log  x'  = 


a„  —  a, 


-  +  3. 


Therefore,  the  corrected  value  is 

log  a:' =  0.242760, 
and  iVoni  this  we  derive 

log  .r"  =  0.237681. 

Those  results  satisfy  the  equations  for  x'  and  x",  and  give 


2'  =  27°  21'    1".2, 
2"  =  29     3  12  .9, 


logr'  =0.294242, 
log  r"  =  0.296087. 


To  find  the  curtate  distances  for  the  first  and  second  observations, 
the  formulte  are 


R  sin  (z'  +  4')  ,„,  ^ 

0    =  ; ; cos  if 


C08,T, 


.■:  =  Kli't!£+j:},^,r, 


am  2  su)  z 

which  give 

log  p'  =  0.133474,        ■      log  p"  =  0.289918. 

Then,  by  means  of  the  equations 


300 


THEORETICAL    ASTRONOMY. 


r'coHh'vG'iiC-Q')     ==t>'cos(A'-Q')-R', 
r'  cos  b'  sin  (f  —  ©')     =  />'  sin  ( A'  —  ©'), 
»•'  gin  h'  =  //  tan  ,5', 

r"  cos  6"  cos  {/"  -  O")  ^  //'  cos  (X"  —  ©")  —  /?", 
r"  cos  6"  sin  (/"  -  ©")  =  f>"  sin  (>■"  -  ©"), 
r"8in6"  =^/'tan,j", 

wo  find  the  following  heliocentric  places : 

r  =  37°  3')'  2()".4,  log  tan  //  ==  8.182H(il„,  log  r'  =  0.2fl4l>4:i 

r  =.  58    58  15  .3,  log  tan  b"  =  8.634209„,  log  r"  =  0.2it«i(l^7. 

The  agreement  of  these  values  of  log  /•'  and  log  >•"  with  tho.se  obtaimd 
directly  from  x'  and  x"  is  a  partial  proof  of  the  numerical  calcula- 
tion. 

From  the  equations 

tan  i  sin  (A  d"  -f  /')  —  R  )  =  ^  (tan  b"  -\-  tan  6')  sec  S  (T  —  T), 
tan  i  cos  (A  {I"  +  /')  —  Q)  ==  a  Uii"  ^"  —  tan6')  cosecA  {F'  —  /'», 
tan(/'—  ft') 
cos  t 


tan  It' 
■we  obtain 


„      tan (r- ft) 
tan  II   = 


SI  =  20(5°  42'  24".0, 
n'  -=  190    55     6  .G 


cost 


i    ^     4°  36'  47".2, 
«"=212    20  53  .5. 


Then,  from 


-"=1^(^  +  1)' 


ft 


n 


n"P\ 
:  n'P", 


we  get 


log  n"  =  9.806832, 
logn'  =9.825408, 


log  ft    =9.653048, 
logft"'  =  9.633171, 


and  the  equations 

r  sin  ((ft'  -  ft)  +  ^  (ft"  -  «'))        =  I-i-"—  sin  i  («"  -  «'), 

r  cos  ((«'  —  ft)  +  i  (ft"  —  «'))        =  — cos  ^  (ft"  —  ft'), 

r"  +  n'r' 


r'"  sin  ((ft'"  —  ft")  +  ^  (ft"  -  ft'))  = 
r'"  cos  ((ft'"  -  ft")  +  ^i  (m"  -  «'))  = 


/'  —  n'r' 


n" 


sin  A  («"  —  ft'), 
cos^(jt  — «;, 


NUMERICAL    EXAMPLK. 


301 


give 


logr    ^.0.308:570, 
lo}r/-"'.^0.:51727:{, 


V      =  100''  :{0'  'u'fy, 

u'"  =  244    5U  32  .o. 


Next,  by  moans  of  the  fonmilii; 


tan  b 
tan  b' 


tun  (I  —  SI)    =  cos  /  tan  u, 

tan  (/'"  —  ft )  =^  cos  /  tan  u'", 

P  cos(A  —  O )  =  r  cos  6  cos  (/  —  ©)  +  Ji, 

f>  mi  (A  —  O  )  =r  con  b  sin  (/  —  O), 

/>  tan  /?  =  r  .sin  i  ; 


tanisinC/ —  ft), 
tan  t. sin  (T— ft), 


cos  (r  —  ©'") 

sin(r'-0"') 


f'"  tan  ,J 


r"'cos6"'cos(r- 
>•'"  cos  6'"  sin  (/'"  - 


0"')  +  R" 
■  ©'"), 


'"  sin  b' 


we  obtain 


/  =  7°  10'  ol".8, 
b  =  +  1  32  14  .4, 
X  =  16  59  y  .0, 
/9  =  +  2  o6  40  .1, 
log/>  =  0.02.5707, 


r  =       91°  37'  40".0,. 
b'"  =  -    4    10  47  .4, 
/"  =      75    23  46  .9, 
fi"'  =  —    3      4  43  .4, 
log//"  =:  0.449258. 


The  value  of  ^'"  thu.s  obtained  agrees  exactly  with  that  given  by 
observation,  but  A  differs  ()".4  from  the  observed  vahie.  This  ditler- 
onco  (Iocs  not  cxeeeil  what  may  be  attributed  to  the  unavoidable 
errors  of  calculation  with  logarithms  of  six  decimal  places.  The 
(littorences  between  the  comi)uted  and  the  observed  values  of  ,9  and 
fi"  show  that  the  position  of  the  plane  of  the  orbit,  as  deterinincd 
l)v  means  of  the  second  and  third  places,  will  not  completely  satisfy 
the  extreme  places. 

The  four  curtate  distances  which  are  thus  obtained  enable  us,  in 
tlie  case  of  an  orbit  entirely  unknown,  to  complete  the  correction  for 
aberration  according  to  the  cijuations  (40). 

The  calculation  of  the  quantities  which  are  independent  of  P', 
P",  (/,  and  Q",  and  which  are  therefore  the  same  in  the  successive 

livpotheses,  should    be   performed   as   accuratelv  as  [)ossible.     The 

c' 
value  of  ~,  required   in   finding  x"  from  x',  may  be  computed 

directly  from 


f  ~       h'^  h'' 


d' 


the  values  of  yi  »"<!  tt  l>ciug  found  by  means  of  the  equations  (29) ; 


302 


TIIEOHKTICA I.   AHTIIONOMY. 


uiul  a  similar  iiu'tluul  may  I »e  adopted  in  the  carto  of   *•     FiirtliiT, 

in  t\w  ('omputation  (tf  x'  and  .r",  it  may  in  some  (u^c's  l)c  advisalilo 
to  employ  one  or  l)otli  of  the  efjiiations  (22)  for  t\w.  Miial  trial.  'I'liii-, 
in  tl)C  i)r('sent  case,  .r"  is  fomid  from  the  first  of  ('(jnations  (27)  liv 
means  of  the  difference  of  two  larj^er  nundwrs,  aiivl  an  error  in  tiie 
last  decimal  place  of  the  logarithm  of  eitiier  of  these  ninnhers  ulUcts 
in  a  f'reater  <lej^ree  the  result  obtained.     Jiut  as  soon  as  r"  is  known 

so  nearly  that  the  logarithm  of  the  factor  1  -|-   „^  reniains  nncliaii<;nl, 

the  second  of  e<piations  (22)  gives  the  value  of  .r"  by  means  of  tin; 
sum  of  two  smaller  numbers.  In  general,  when  two  or  more  I'or- 
nnihe  for  iinding  tiie  sune  (piantity  are  given,  of  those  which  nro 
otherwise  ecpially  accurate  and  convenient  tor  logarithmic  calculation, 
that  in  which  the  number  sought  is  obtained  from  the  sum  of  smaller 
inimbers  should  be  preferred  instead  of  that  in  which  it  is  obtaintii 
by  taking  the  difference  of  larger  numbers. 

The  values  of  /•,  /•',  r",  r'",  and  u,  a',  u" ,  n'",  which  result  from 
the  first  hypothesis,  suffice  to  correct  the  assumed  values  of  V,  1'", 
Q',  and  Q".     Thus,  from 


r^k{,t"~t'), 


tan;if  =  \i 


J~> 


tan/' =  4'-, 


n 


sin  y  cos  (r  =:  sin  A  («" —  n'),  sin  y"  cos  G"  =  sin  S  (n' —  u), 

sin  y  sin  G  =  cos  \  {u" —  u' )  cos  2/,    sin  /'  sin  G"  =  cos  \  (u' —  u)  cos  2/", 

cos  J'  =coSo(tt" — n')  sin  2/,     cos/'  =cosA(n' — u)s'm'lx", 


sin/" 

cos 

G"':=sini(«"'- 

-u"), 

sin/" 

sin 

G"'^cosy(u"'- 

u") 

cos  2/" 

cos/" 

=^cos\{u"'- 

■u") 

sin  2/"; 

m 

t'  cos';f 
v''cos''/ 

"'  =   ,-c..V"' 

7tt"'  = 

t""co8«/" 
"r"»cosV"" 

J 

sin"  {y 
cos  ^  ' 

sinM/' 
^  ~  cos/" 

r= 

sin^J/" 
cos/"' 

m 

,'"  = 

m'" 

^ 

~2+J  +  f' 

^  ~'-\-j"+r 

1+/"+^"" 

X 

x'"  = 

8""        ''     ' 

in  connection  with  Tables  XIII.  and  XIV.  we  find  s,  s",  and  h'". 
The  results  are 


Iogr=n0.n7r)0441, 

^=10    42  ")')  .U, 

logm  .  -«.1H<5217, 
logj  --^  7.y4«()!»7, 
log*   -^0.0(Wr,248, 


NUMERICAL   EXAMPLE. 

log  r"-_  0.1880714, 
/'.---  44°  32'    1".4, 
r"-^  15    i;{  45  .0, 
log  wi"      8.51  G727, 
log/'  -=  8.2«5()()i;{, 
log  a"  =^0.0174621, 


303 


log  r'"-^- 

0.1800(541, 

/.'" 

45°  41' 55" 

.2, 

r" 

1(5    22  48 

.5, 

log  m'"  - 

:  8.5}»05!»(5, 

log/"--. 

.  8.325:5(55, 

log  «"'=^ 

:  0.02040(53. 

Tlicn,  l)y  means  of  tlie  forinuhc 

r  J< 

P' 

Of 


t" 


TT 


•  M,i"    rr"  cos  \  («"  —  u')  cos  .J  («"  —  «)  cos  J  (u'  —  n)' 


P"  = 


t"'" 


« 


TT 

^' ""  -  «*.'"  '  ?r"'  cos  A  («" 


r"» 


ri' 


we  obtain 


logP'  ^9.8402100, 
log  P"=.  9.8077(515, 


u")  cos  ^  («'"  —  «')  cos  ^  («"  —  u')' 

log  (^  ==  9.840753G, 
log  (}'  =  9.8824728, 


with  which  the  next  approximation  may  be  completed. 

We  now  recompute  cj,  oJ',f',f"y  x',  x",  &c.  precisely  as  already 
illustrated;  and  the  results  are 


L)gc;=.  0.5413485,., 
log/'  =  0.047(>(>14„, 
logs/  ==0.2427528, 

g'  =  27°  21'  2".71, 
log  r' =0.2942369, 
log  ^' =0.1334635, 
log»i  =9.6530445, 
log  h' =9.8254092, 

Then  we  obtain 


logc„"  =  9.8076649,., 
log/"  =9.8893851, 
log  a/'  =0.2376752, 

2"  =  29°  3' 14"..., 
logr"  =0.2960826, 
log^"  =0.2899124, 
log  n"  =  9.8068345, 
log7t'"  =  9.6331707. 


V  =  37°  35'  27".88, 
r'=58    58  16  .48, 


log  tan  b'  =  8.1828572„, 
logtani"=8.6342073„. 


log/  =0.2942369, 
logr"  =0.2960827. 


These  results  for  log  J*'  and  logr"  agree  with  those  obtained  directly 
from  s'  and  z",  thus  checking  the  calculation  of  '^'  and  i]/"  and  of 
the  heliocentric  places. 
Next,  we  derive 


ft  =  206°  42'  25".89, 
n'  =  190    55    6  .27, 


i  =     4°  36'  47".20, 
«"  =  212    20  52  .96, 


304 


TIIKOFIKTICAI,   ASTUONOMV. 


niul  from  u" -—  n',  r',  r'\  »,  n",  n\  and  u"\  we  ohtnin 

logr   =^  o.:1()m;{7;{4,  «     .  i(i()° ;{()'  :):)".4"), 

log r"'r^O.:n 72(174,  n"'.-244    ')!>  :M  m. 

For  the  purpose  of  provinjj  the  accuracv  of  the  minioriciil  rostilt>i, 
wc  compiiU!  also,  tis  in  the  first  approximation, 


/=        7°  KJ'  ^A"^A, 

6=+    1    ;{2  14  .07, 

•     A  =       IG    oj)     9  .:{8, 

/9  =  +    2    m  39  .04, 

lojr/>=:().()2.")(}!)(;0. 


/'"=  01°  37'  41".20, 
6'"=—  4  10  47  .:{({, 
r'=  7.")  23  4(>  .99, 
/9"'=:—  3  4  43  .33, 
log//"    -  0.44925,39. 


The  values  of  /.  and  I'"  thus  found  differ,  respectively,  only  0".0| 
and  0".09  from  tiiose  given  l)y  the  normal  places,  and  hence  the 
accuracy  of  the  entire  calculation,  hoth  of  the  (piantities  which  arc 
independent  of  /",  P",  Q',  and  Q'\  and  of  those  which  depend  (in 
the  successive  hypotheses,  i.s  completely  proved.  Thi.s  condition, 
however,  nmst  alway.s  be  satisfied  whatever  may  be  the  twsunieJ 
values  of  P',  P",  q\  and  (}» . 
From  /•,  r',  »,  «',  &c.,  we  derive 

log  «  =-  0.0080254,        log  «"  =  0.0174637,        log  s'"  =  0.0204076, 

and  hence  the  corrected  values  of  P',  P",  Q',  and  Q"  become 


logP': 
logP": 


9.8462110, 
:  9.8077622, 


log  (/  =  9.8407524, 
log  (/'  =  9.8824726. 


These  values  differ  so  little  from  those  for  the  second  approximation, 
the  intervals  of  time  between  the  observations  being  very  large,  that 
a  further  repetition  of  the  calculation  is  unnecessary,  since  the  results 
which  would  thus  be  obtained  can  differ  but  .slightly  from  those 
which  have  been  derived.  We  shall,  therefore,  complete  the  deter- 
mination of  the  elements  of  the  orbit,  using  the  extreme  places. 
Thus,  from 

To  ==  k  it'"  ^  0,  tan  xo  =  \  — ' 

sin  y-j  cos  Ga  =  sin  ^  (v!"  —  u), 

sin  Yo  sin  ^a  =  cos  ^  («'"  —  \C)  cos  2xq, 

cos  Y^  =  cos  ^  (tt'"  —  u)  sin  2/^, 


«Jn  = 


(r  +  r'")' cosVo' 


Wo 


_  sin"  \Ya_ 
-'"        cosYo  ' 

^0  —  "7T      Jv 


NUMEUICAL    KXAMl'LE. 


30o 


wc  get 


r.r   4-2'  14' :{()".  17, 


logtiiii  ^'„     M.o.v.M !»:.:?__, 

l(.^,'»;i„       !t."]7!MC.'»;, 
loj,'.i-„       M.iHl(»M:!!»7. 


I'lii'  ((iriniilii 


givw 


/.„n"'»'inrH"'— u)\» 


)' 


lojry,     o.;{7l'.Mni; 

and  if  we  comimtc  t\\v.  sniue  (jiiuiitity  i)y  iiU'IIIIm  of* 

/  xr'r"^\i\  ( >("—  ,i' )  \'     I  V'/Y'«in  ( »'  -  » )  \''     /  .■<"' r" ,'"  An ( .t"'-»/'  >  v' 
?'^(     --      r )       \  r"  )^^\ r"'  )' 

till'  sopanito  results  aro,  rospoctivcly,  (>..'J712'"5!>7,  (  .•')7r2 1 IH,  and 
0.'"{7r24l4.  The  clitKTeiices  hetweeii  these  results  nre  ^ery  siimll,  uii*l 
arise  hoth  from  the  iiiiav(>i<lal)le  errors  of  ealeiilation  aiid  from  tin 
ijeviation  of  the  adopted  values  of  /",  J"',  /,  and  (/'  from  the 
limit  of  ueeiiruey  attainable  with  logarithms  of  s"\'v]\  decimal  places. 
A  vnriatioii  of  only  (>".li  in  the  values  of  ii'  a  and  a'"  -  n"  will 
pn  hice  an  entire  ueeordunce  of  the  particular  results. 
J''rom  the  equation.s 


acosv> 


sin  .J(h"'—  It) 
sin  ((£'"-  E) 


V'rr 


QOitp 


a  cos  v> 


we  ol)taui 


(£'"-£)  =  17°  35' 42".l 2, 


logfrt  cos  9')  ^  0.37!)G883, 


The  formulic 


logcosic^r^O.OinorjlH. 


i'An{u>  .-^,(«"'-f-«)) 


P 


cos  Yo  1    '■>•' 


rr::^^  tan  G, 


0) 


e  cos(w  —  tJ  (n"'  +  n))  = -—  --  —  sec  .J  («"'  —  u) 


ffive 


oj  =  197°  38'  8".48, 


V 


11°  lo'52".2: 


cos  yo  V  j*r"' 


log  e  =  log  sin  <p  =  9.2907881 , 
r  =.  «  +  J2  =  44°  20'  34".37. 


This  result  for  <p  gives  ogcos^  —  9.991o521,  which  differs  only  3 
ill  the  last  decimal  place  from  the  value  found  from  ^)  and  acosf. 
Thou,  from 

30 


TlIEOnETICAL  ASTRONOMY. 


P 


3  I 


the  vahic  of  k  being  expressed  in  seconds  of  arc,  or  log^  ==  3.55000G6, 
•\vc  get 

log  a  =  0.;38813">9,  log^  =  2.9678027. 

For  the  eccentric  anomalies:'  Ave  liave 

tan  IE  ^  tan  I  ^u  -—  w)  tan  (45°  —  ^^), 
hill  IE'  =.:tanU«'  —  w)  tan  (4-5°  —  I^?), 
tan  5 E"  -.-:  tan ]  (u"  —  w)  tan (4o°  —  l,p), 
tan  Ie'"  =^  tan  l  («'"—  w)  tan  (4o°  —  l-p), 


from  which  the  results  are 

E  =329°  11'46".01, 
£'=:354    29  11  .84, 


E"  =-^2°    5'33".63, 
E'"  =  39   34  34  .65. 


The  value  of  }[E"'  —  E)  thus  derived  differs  only  0".03  from  tliat 
obtained  directly  from  Xq. 

For  the  moan  anomalies,  we  have 


M  =rE~c  sin  E, 
M'^E'  —  esinE', 


M"  =E"-e  sin  E", 
M"'^E"'-esinE"', 


which  give 


M  =  334°  55'  39".32, 

M"  ^ 

-   9°  44'  52".82 

M'  =  355    33  42  .97, 

M'"  - 

-  32    26  44  .74, 

Finally,  if  3/„  denotes  the  mean  anomaly  for  the  ejioch  T=  1864 
Jan.  1.0  nicaii  time  at  Greenwich,  from 


M^r=M-!i{t  —  T)     =  M'  —  ii{i!  - 
=  iV"  -  //  ( V  -T)  =  M'"  -  II  (r  • 


T) 


T), 


we  obtain  the  four  values 


il/o  =  1°  29'  39".40 
39  .49 
39  .40 
39  .40, 


the  agreement  of  which  completely  proves  the  entire  calculation  of 
the  elements  from  the  data.  Collecting  together  the  several  results, 
we  have  the  following  elements : 


NUMERICAL    EXAMPLE. 


307 


Epoch  =^  1864  Jan.  1.0  Groonwicli  mean  time. 
JI/=     1°  29'  3i»".42 


r  :^    44    20  3?  .37 
SI  =  20(5    42  25  .89 


I    pA'lijjtic  and  Mean 
E(iuinox  18G4.0. 


i=.     4    3«  47  .20  j 
<f=    11    1.-)  .-)2  .22 

los?a  =  0.3881359 

logM=:  2.9078027 
/x  ==  928".54447. 


102,  The  elements  thu.s  derived  completely  represent  the  four  ob- 
served longitudes  and  tiie  latitudes  for  the  second  and  third  places, 
w'.iich  are  the  actual  data  of  ilio.  })rol»lem  ;  hut  for  the  extreme  lati- 
tudes the  rjsiduids  are,  computation  minus  observation, 


A,3  =  —  4".47, 


A  5'"  : 


+  1".23. 


These  I'emaining  errors  arise  chiefly  from  the  circumstance  that  tlie 
position  of  the  j)ianc  of  the  orbit  cannot  be  determined  from  the 
second  and  third  places  with  the  same  degree  of  precision  as  from 
the  extreme  places.  It  would  be  advisable  therefore,  in  the  final 
ai)proximation,  as  soon  as  ft',  f)",  n,  n",  )i',  and  /('"  are  obtained,  to 
compute  from  these  and  the  data  furnislied  directly  by  observation 
the  curtate  distances  for  the  extreme  places.  The  corresj)onding 
heliocentric  places  may  then  be  found,  and  hence  the  position  of  the 
plane  of  the  orbit  as  determined  by  the  first  and  fourth  observations. 
Thus,  by  means  of  the  equations  (37)  and  (38),  we  obtain 


log  ^  =  0.0256953, 


log/)' 


0.4492542. 


With  these  values  of  p  and  p'",  the  following  heliocentric  places  are 
obtained : 

I  ^  7°1G'51".54,        log  tan  6    =-8.4289004,        log/-    =0.3083732, 
r=:91   37  40  .90,        log  tan  6'"  ==  8.8G38549„,      log?-'"  =  0.3172078. 

Then  from 

tan  /  .«in  (A  (/'"  +  0  -  SX)  =  i  (tan  b'"  +  tan  b)  sec  A  (r  -  I), 
tan  i  cos  (l  (J!"  +  0  —  Si  )  =  2  (tan  b'"  —  tan  b)  cosec  A  {l'"  —  I), 

we  get 

SI  =  200°  42'  46".23,  i  =  4°  36'  49".76. 

For  the  arguments  of  the  latitude  the  results  are 

u  =  160°  30'   35".99,  «'"  =  244°  59'  12".53. 


308 


TIIEORETICAI.   ASTRONOMY. 


Tlic  equations 


tan  //  =  tan  /  sin  (/'  —  £1), 
tan  b"  ^  tan  i sin  H"  —  Q), 


give 


logtani'  =  8.1827129„, 


log  tan  b"  ^  8.6342104. . 


and  the  comparison  of  these  results  with  those  derived  diroetly  from 
//  and  o"  exhibits  a  dit!erenee  of  +  1".04  in  b',  and  of  —  0".0G  i„ 
//'.  Hence,  the  jiosition  of  the  plane  of  the  orbit  as  determined  from 
the  extreme  ])laces  very  nearly  satisfies  the  intermediate  latitudes. 

W  we  comj)ute  the  remainin»;  elements  by  means  of  these  values 
of  r,  !•'",  and  )«,  a'",  the  separate  results  are : 


lojr  tan  r;„  ^.  8.0r)22282„, 
log.V  =  0.2017731, 
logj;--- 0.3712405, 
log  (a  008  <f)  =-  0.37!K)884, 

w  :=  107°  37'  47".72, 

^==    11    15  52  .46, 

log  n  ==:  0.3M81365, 

£-:329°  11'47".24, 

jI/=3.34    55  40  .46, 

Mo=      1    ':9  40  .36, 

Hence,  the  elements  are  as  follows ; 


log  «i„  =  0.7170026, 
log  .!•„  =  8.0608307, 
J(J5;"  — £)  =  17°35'42".12, 
log  cos  ^jr=<).t)0]  5521, 
log  e  =^  0.2007006, 
log  cos  v'  =  0.00 1.5520, 
log/i  =  2.0678010, 
/;"'  =  30°  34'  35".70, 
J/'"  =  32    26  45  .49, 


3/„ 


1    29  40  .37. 


Ecliptic  and  INFean 
Equinox  1864.0. 


Epoch  =^  18()4  Jan.  1.0  Greenwich  mean  time 
i/--      1°  20'  40".36 
X  =   44    20  32  .05 
Si  =  206    42  45  .23 
i  =     4    36  40  .76 
P=    11     15  52  .46 
logo  =  0.3881305 
//  =^  92«".5427. 

It  appears,  thorefon",  that  the  principal  effect  of  neglecting  the 
extreme  latitudes  in  the  determination  of  an  orbit  from  four  obser- 
vations is  on  the  in<'lination  of  the  orbit  and  on  the  lontjitude  of  the 
a.scending  node,  the  other  elements  being  very  slightly  changed.  The 
elements  thus  derived  represent  the  extreme  places  exactly,  and  if 
we  eom])ute  the  second  and  third  places  directly  from  these  elements, 
we  obtain 


M'  :=  355°  33'  43".88, 
j;'=354  29  12  .93, 
v'    =  353    16  59  .07, 


M"=  9°44'53".73, 
£"  =  12  5  34  .81, 
v"    =14   42  45  .96, 


NUMEKICAL    EXAMPI-E. 


309 


logr'  =  0.2n42.%6, 

It'  :=       190°  54'  46".79, 


V 


j7    85  27  .75, 


b'  =  —     0    52  21  .25, 

A'=         10    14  17  .35, 

/5'=—      1    15  47  .67, 
log// =.  0.1334(334, 


log  >•"  =  0.2000826, 

u"  =^       212°  20'  33".(;8, 

r==         58    58  10  .50, 

h"--=—      2    27  50  .00, 

/'=         29    53  21  .99, 

/?■'  =  —      2    29  57  .02, 

log  |o"=.  0.2899122. 


Honco,  the  residuals  for  the  second  and  third  places  of  the  planet 

a«! — 

Comp.  —  Obs'. 
aa'  :=  —  0".22,  A,j'  =  +  1".53, 

aA"  =       0  .00,  A,J"  =^  —  0  .00 ; 

and  the  elements  very  nearly  represent  the  four  normal  places.  Since 
the  interval  between  tlie  extreme  places  is  223  days,  these  elements 
imi?t  represent,  within  the  limits  of  the  errors  of  observation,  tlio 
entire  series  of  olx^fcrvations  on  which  the  normals  are  based.  It 
may  be  observed,  also,  that  the  successive  approximations,  in  the 
case  of  intervals  which  are  very  large,  do  not  converge  with  the 
same  degree  of  rapidity  as  Avhcn  the  intervals  are  small,  and  that  in 
such  cases  the  numorifal  calculation  is  very  much  abbreviated  by  the 
determination,  in  the  tirst  instance,  of  the  assumed  values  of  P' ,  P", 
Q',  and  Q"  by  means  of  approxinuite  elements  already  known.  For 
the  first  determination  of  an  unknown  orbit,  the  intervals  will  gene- 
rally be  so  small  that  the  first  assumed  values  of  these  quantities,  as 
determined  by  the  equations 


P 
P 


—  ^»i  \^       a       ,"3      / »  V   —  2  ■  •    > 


Mill  not  differ  much  from  the  correct  values,  and  two  or  three 
liv|)otlieses,  or  even  less,  will  be  sufficient.  But  when  the  intervals 
are  large,  and  especially  if  the  eccentricity  is  also  considerable,  several 
hypotheses  may  be  required,  the  last  of  which  will  be  facilitated  by 
using  the  equations  (82)^. 

The  application  of  the  formula;  for  the  determination  of  an  orbit 
from  four  obser\ations,  is  not  confined  to  orbits  whose  inclination  to 
the  ecliptic  is  very  small,  corresponding  to  the  cases  in  which  the 
method  of  finding  the  elements  by  means  of  three  observations  fails, 


4 


i 


310 


THEORETICAL   ASTHOXOMY. 


or  at  loa.st  becomes  very  uncertain.  On  tlie  contrary,  tlie.se  formula; 
apply  cipiall}  well  in  the  case  of  orbits  of  any  inclination  whatever, 
and  .since  the  labor  of  com[)Utin}^  an  orbit  from  four  observations 
does  not  much  ex(;eed  that  required  when  only  three;  observed  places 
are  used,  while  the  results  must  evidently  be  more  approximate,  it 
will  be  cxpei'ient,  in  very  many  cases,  to  use  the  formulai  given  in 
this  cha[)tcr  both  for  the  first  approximation  to  an  unknown  orbit 
and  for  the  subsequent  determination  from  more  complete  data. 


CIRCULAR  ORBIT. 


311 


CHAPTER  YI. 


ISVmTIOATIOS  OP  VARIOUS  FORMl'L.K  FOIl  TI[K  COURECTIOX  OK  THE  APPUOXIMATE 
ELEMENTS   OF   THE   ORUIT   OF    A    UEAVEXLY    UODV. 

103.  In  the  case  of  the  discovery  of  a  planet,  it  is  often  couvc- 
niont,  before  sufficient  data  have  been  obtained  for  the  determination 
of  elliptic  elements,  to  compute  a  system  of  circular  elements,  an 
cphemeris  computed  from  these  being  sufficient  to  follow  the  planet 
for  a  brief  period,  and  to  identity  the  comparison  stars  used  in  dif- 
ferential observations.  For  this  purpose,  only  two  observed  places 
are  required,  there  beinu;  l)ut  four  elements  to  be  determined,  namely, 
J2,  i,  (I,  JUid,  for  any  instant,  the  longitude  in  the  orbit.  As  soon  as 
a  has  been  found,  the  geocentric  distances  of  the  planet  for  tiic 
instants  of  observation  may  be  obtained  by  means  of  the  formuhc 


A  —R  cos  i   4-  v/a'  —  R'  siii'^  4, 
J"  =  ii"co3V'+V'a» 


R"'mi'^", 


(1) 


the  values  of  -v//  and  ij/"  being  computed  from  the  ecpiations  (42),,  and 
(43)3.  For  convenient  logarithmic  calculation,  we  may  first  find  z 
and  z"  from 

Rsin^  .     „      ^"sinV 


smz 


a 


sm  z 


since  the  formula)  will  generally  be  required  for  cases  such  that  these 
angles  may  be  olv-ained  with  sufficient  accuracy  by  means  of  their 
sines.     Then  we  have 


i?  sin  (2 +  4)        5 
p  = :j:rz ^^^  '^> 


f   = T—r, cos ,i  ,       (3) 


sm  z  sni  z 

from  which  to  find  p  and  p".     These  having  been  found,  we  have 

jTJsinfA  —  O) 


tan(/—  O)  = 
sin  6  =^ 


ft  tan  (3 


(-1) 


for  the  determination  of  I  and  b,  and  similarly  for  /"  and  b".     The 


if-! 


;n2 


THEOUETlCAIi    ASTUONOMY. 


inclination  of  tlic  orbit  and  the  lonf^itude  of  tlic  ascending  node  are 
then  lound  by  means  of  tlie  forniulic  (7o);„  and  the  ar}j;ninents  of  tiie 
latitndc  by  means  of  (77);i.  Since  u''  —  n  is  the  distance  on  the  celes- 
tial sphere  between  two  points  of  which  the  heliocentric  spherical 
co-ordinates  are  /,  6,  and  /",  6",  we  have,  also,  the  equations 

sin  («"  —  tO  sin  B  =  cos  b"  sin  (l"  —  I), 

sin  (u"  —  u)  cos  B  =  cos  b  sin  b"  —  sin  b  cos  b"  cos  (I"  —  /)> 

cos  («"  —  It)  =:  sin  b  sin  b"  +  cos  b  cos  b"  cos  (l"  —  /), 

for  the  determination  of  »" — n,  tiie  angle  opposite  the  side  90°  — />" 
of  the  spherical  triangle  being  denoted  by  />.  The  solution  of  the;<e 
('(piations  is  fa(  ilitated  by  the  introduction  of  auxiliary  angles,  as 
already  illustra.ed  for  similar  cases. 

[n  a  circular  orbit,  the  eccentricity  being  equal  to  zero,  u"  —  u 
expresses  the  mean  motion  of  the  planet  during  the  interval  t"—t, 
and  we  must  also  have 


.1 


t"-t^    r'(it"~u), 


(5) 


the  value  of  k  being  expressed  in  seconds  of  arc,  or  log  k  —  3.55000(30. 
These  formuhe  will  be  apj)licd  only  when  the  interval  t" — (  is 
small,  and  for  the  case  of  the  asteroid  planets  we  may  first  assume 

a  =  2.7, 

which  is  about  the  average  mean  distance  of  the  group.  With  this 
we  compute  f>  and  (>"  by  means  of  the  equations  (2)  and  (3),  and  the 
corresponding  heliocentric  places  by  means  of  (4).  If  the  inclination 
is  small,  u" —  (t  will  (litter  very  little  from  /"  —  /,  Therefore,  in  the 
first  approximation,  when  the  heliocentric  longitudes  have  been  found, 
the  corresponding  value  of  t" — t  may  be  obtained  from  equation  (5), 
writing  /" — /  in  place  of  n" — u.  If  this  comes  out  less  than  the 
actual  interval  between  the  times  of  observation,  we  infer  that  the 
assumed  value  of  a  is  too  small ;  but  if  it  comes  out  greater,  the 
assumed  value  of  «  is  too  large.  The  value  to  be  used  in  a  repetition 
of  the  calculation  may  be  computed  from  the  expression 

log  a  =  i  (log  it"  -t)  +  log  k  -  log  («"  -  «)), 

the  dif!erenee  u" —  n  being  expressed  in  seconds  of  arc.  With  this 
we  recompute  p,  ft",  f,  and  /",  and  find  also  b,  b",  ft,  i,  u,  and  u". 
Then,  if  the  value  of  a  computed  from  the  last  result  for  m"—  u 
differs  from  the  last  assumed  value,  a  further  repetition  of  the  calcu- 


CIRCULAR   ORIIIT. 


813 


latioii  1)oronios  necossaiy.  But  whoa  three  snocossivo  a])i)roximate 
values  of  <t  liavo  been  found,  the  corroct  value  may  !>(>  readily  inter- 
polated aeeordinjj;  to  the  proeess  already  illustrated  for  similar  eases. 

As  soon  as  the  vahie  of  a  has  heen  obtained  whieli  eoni|)letely 
gatisfics  e(|uation  (o),  this  result  and  the  eorrespondin»;  values  of  SI, 
i,  and  the  ar<>;uinent  of  the  latitude  for  u  fixed  epoch,  complete  the 
system  of  eireidar  elements  whieh  will  exactly  satisfy  the  two  observed 
places.  li'  we  (Ujiiote  by  ii^  the  ar<fumcnt  of  the  latitude  for  tiie  epoch 
T,  we  shall  have,  for  any  instant  t, 

u  being  tlie  mean  or  actual  daily  motion  computed  from 

k 


li 


ai 


The  value  of  n.  thus  found,  and  r  =-  a,  substituted  in  the  formulse  for 
computing  the  places  of  a  heavenly  body,  will  furnish  the  approxi- 
mate ephemeris  required. 

The  corrections  for  parallax  and  aberration  are  neglected  in  the 
first  determination  of  circular  elements;  but  as  soon  as  these  approxi- 
mate elements  have  been  derived,  the  geocentric  distances  may  be 
ooiiiputed  to  a  degree  of  accuracy  suHicient  for  ap|)lyiiig  these  <'or- 
rcctions  directly  to  the  observed  places,  preparatory  to  the  determi- 
nation of  elliptic  elements.  The  assumption  of  r'^^  «  will  also  be 
sufficient  to  take  into  account  the  term  of  the  second  order  in  the  first 
assumed  value  of  P,  according  to  the  first  of  equations  (98)^. 

104.  Wiien  approximate  elements  of  the  orbit  of  a  heavenly  body 
have  been  determined,  and  it  is  desired  to  correct  them  so  as  to  satisfy 
as  nearly  as  possil)le  a  series  of  observations  including  a  much  longer 
interval  of  time  than  in  the  case  of  the  observations  used  in  finding 
those  a])proximate  elements,  a  var'cty  of  methods  may  be  aj)plied. 
For  a  very  long  series  of  observations,  the  approximate  elements 
being  such  that  the  squares  of  the  corrections  which  must  be  ajjplied 
to  them  may  be  neglected,  the  most  complete  method  is  to  form  the 
equations  for  the  variations  of  any  two  spherical  co-ordinates  which 
fix  the  place  of  the  body  in  terms  of  the  variations  of  the  six  ele- 
nienis  of  the  orbit;  and  the  differences  between  the  computed  places 
for  ditferent  dates  and  the  corresponding  observed  places  thus  furnish 
equations  of  condition,  the  solution  of  which  gives  the  (jorrections  to 
be  applied  to  the  elements.     But  when  the  observations  do  not  in- 


! 


i 


814 


TIIKORETIf'AI.    ASTRONOMY. 


cliulf!  a  very  loiipj  interval  of  time,  instead  of  forminfc  the  er|uations 
for  the  variations  of  the  geocentrie  places  in  terms  of  tlie  varial ictus 
of  the  elements  of  the  orbit,  it  will  be  more  convenient  to  form  the 
equations  for  these  variations  in  terms  of  (inantities,  less  in  nnniliir, 
from  which  the  elements  tluimsclvcs  are  readily  obtained.  If  no  as- 
sumption is  made  in  regard  to  the  form  of  the  orbit,  the  quantities 
■which  present  the  least  diificulties  in  the  numerical  calculation  arc 
the  geocentric  distances  of  the  body  for  the  dates  of  the  cxtrtiiu' 
observatifms,  or  at  least  for  the  dates  of  those  which  are  best  adapted 
to  the  determination  of  the  elements.  As  soon  as  these  distances  are 
accurately  known,  the  two  corresponding  complete  observations  are 
sufficient  to  determine  all  the  elements  of  the  orbit. 

The  ap])roximate  elements  enable  us  to  assume,  for  the  dates  t  and 
t",  the  values  of  J  and  J";  and  the  elements  computed  from  those 
by  means  of  the  data  furnished  by  observation,  will  exactly  represent 
the  two  observed  places  employed.  Further,  the  elements  may  )«? 
supposed  to  be  already  known  to  such  a  degree  of  approximation  that 
the  squares  and  products  of  the  corrections  to  be  applietl  to  the 
assumed  values  of  J  and  J"  may  be  neglecte<l,  so  that  we  shall  have, 
for  any  date, 

da        ,    ,  ^    da 


COS  0  Aa 


COS  ')■    .  ,   A  J  -f  cos  ^    "  ,„   A  J", 


A^=r 


A  J  4- 

dJ      ^ 


do 


(6) 


rfJ 


77  A  J". 


If,  therefore,  we  compare  the  elements  computed  from  J  and  J"  with 
any  number  of  additional  or  intermediate  observed  places,  each  ob- 
served spherical  co-ordinate  Avill  furnish  an  equation  of  condition  for 
the  correction  of  the  assumed  distances.  But  in  order  that  the  equa- 
tions (G)  may  be  applied,  the  numerical  values  of  the  partial  ditferen- 
tial  coefficients  of  a  and  d  with  respect  to  J  and  J"  must  be  found. 
Oidinarily,  the  best  method  of  effecting  the  determination  of  tliese  is 
to  compute  three  systems  of  elements,  the  first  from  J  and  J",  the 
second  frotn  J  +  i)  and  J",  and  the  third  from  J  and  J"  +  D",  D 
and  B"  being  small  increments  assigned  io  J  and  J"  respectively. 
If  now,  for  any  date  t',  Ave  compute  a'  and  o'  from  each  system  of 
elements  thus  obtained,  we  may  find  the  values  of  the  differential 
coefficients  sought.  Thus,  let  the  spherical  co-ordinates  for  the  time 
t'  computed  from  die  first  system  be  denoted  by  a'  and  o';  those 
computed  from  the  second  system  of  elements,  by  a'  -f  «  -^co  "'  '''"'^ 
8'-\-  d:  and  those  from  the  third  system,  by  a'+  a"  sec  d'  and  o'--  *'"• 
Then  we  shall  have 


VARIATION   OF   TWO   GEOCENTIUC    DISTANCES. 


310 


fla 

a 

rfJ 

-D' 

ila' 

«" 

dJ" 

~  D" 

cos  o  -— -   = 


C08  'J 


ami  the  cqiiations  (G)  give 


d>t  _  d 

dJ  "  />' 

(/'J  _  d" 

dJ"""]/' 


cos  '5'  Ao' 


I)     ^  JJ"     ' 

^.        d       ,    ,    d"       ,,, 
^0=  j^^J  +  ^.,^J". 


(7) 


(8) 


111  tlio  «ime  manner,  computing  the  plaeos  for  various  dates,  for 
which  observed  places  are  given.  In*  means  of  each  of  tlie  three  systems 
of  elements,  the  equations  for  the  correction  of  J  and  J",  as  deter- 
mined by  eaeli  of  the  additional  observations  employed,  may  be 
I'onned. 

105.  For  the  purpose  of  illustrating  the  application  of  this  method, 
let  us  suppose  that  three  observed  jtlaces  are  given,  referred  to  the 
eclijttie  as  the  fundamental  plane,  and  that  the  corrections  for  })aralla.\', 
aberration,  precession,  and  nutation  liave  all  been  duly  applied.  By 
means  of  the  approximate  element^j  already  known,  we  compute  tiic 
values  of  J  and  J"  for  the  extreme  places,  and  from  tliese  the  helio- 
centric })laces  arc  obtained  by  means  of  the  equations  (71).j  and  (72).,, 
writing  Jcos/9  and  J"  cos^j'"  in  place  of  ft  and  p".  The  values  of 
Ji,  /,  u,  and  u"  will  be  obtained  by  means  of  the  formuhe  (70).,  and 
(77)3;  and  from  r,  r"  and  u"  —  u  the  remaining  elements  of  the 
orbit  are  determined  as  already  illustrated.  The  first  system  of  ele- 
ments is  thus  obtained.  Then  we  assign  an  increment  to  J,  which 
we  denote  by  D,  and  with  the  geocentric  distances  J  +  D  and  J" 
we  compute  in  precisely  the  same  manner  a  second  system  of  ele- 
ments. Next,  we  assign  to  J"  an  increment  /)",  and  from  J  and 
J"  -~  D"  a  third  system  of  elements  is  derived.  Let  the  geocentric 
longitude  and  latitude  for  the  date  of  the  nuddle  observation  com- 
puted from  the  first  system  of  elements  be  deignated,  respectively, 
by  //  and  /9/ ;  from  the  second  system  of  elements,  by  X^'  ^^'i^  t%'  > 
and  from  the  third  system,  by  A^'  and  fi/.     Then  from 


(9) 


we  compute  a,  a",  d,  and  d",  and  by  means  of  these  and  the  valuea 
of  D  and  D"  we  form  the  equations 


(;.;_a;)oos;V, 

d   -;V-/5/. 

(V-V)eos<v, 

rf"=/v--v. 

316 


TIIEOHETICAL    ASTUOXOMY. 


D       ^  I)" 


0(w,5'aA', 


\i', 


(10) 


for  tlic  <l('torniiiiatif>ii  of  the  convctions  to  be  ni)pli('(l  to  the  first 
assumed  values  of  J  niid  J",  l)y  means  of  the  ditU'renees  b'.'twccii 
observation  and  computation.  The  observed  hmiiitUiU.'  and  hititiidc 
beinj^  (U'lioted  by  //  and  ,'i',  resj)ectively,  we  shall  have 


cos;?' A/.'=;(/ 


(11) 


-//)cos;j', 

for  findin*;  the  values  of  the  seeond  members  of  the  erjuatifuis  (10), 
and  then  bv  elimination  we  obtain  the  values  of  the  eorreetions  iJ 
and  aJ"  to  be  ai)j)lied  to  the  assumed  values  of  the  distance:!. 
Finally,  we  compute  a  fourth  system  of  elements  corresponding  to 
the  geocentric  distances  J  —  a  J  and  J"  +  aJ"  either  directly  from 
these  values,  or  by  interj)olatiou  from  the  three  .systems  of  elements 
already  obtained ;  and,  if  the  first  assumption  is  not  considerably  in 
error,  these  elements  Avill  exactly  represent  the  middle  jilaee.  It 
shoidd  be  observed,  however,  that  if  the  second  system  of  elenuiits 
re])resents  the  middle  place  better  than  the  first  system,  /.„'  and  {ij 
should  be  u.sed  instead  of  //  and  ,9/  in  the  equations  (11),  and,  in 
this  case,  the  final  .system  of  elements  must  be  computed  with  the 
distances  J  -f  i^  +  a  J  and  d"  +  aJ".  Simihirly,  if  the  middle 
place  is  best  represented  by  the  third  system  of  elements,  the  cor- 
rections will  be  obtained  for  the  distances  used  in  the  third  hy- 
pothesis. 

If  the  computation  of  the  middle  place  by  means  of  the  final  olo- 
mcnts  still  exhibits  residuals,  on  account  of  the  neglected  terms  of 
the  second  order,  a  repetition  of  the  calculation  of  the  corrections 
,  aJ  and  aJ",  using  these  residuals  for  the  values  of  the  seeond 
mend)ers  of  the  ecjuations  (10),  will  furni.sh  the  values  of  the  dis- 
tances for  the  extreme  places  with  all  the  precision  desired.  The 
increments  I)  and  D"  to  bo  assigned  successively  to  the  first  assumed 
values  of  J  and  J"  may,  without  difliculty,  be  so  taken  that  the 
true  elements  shall  dificr  but  little  from  one  of  the  three  systems 
computed ;  and  in  all  the  formuhe  it  will  be  convenient  to  use,  in- 
stead of  the  geocentric  distances  themselves,  the  logarithms  of  these 
distances,  and  to  expre.ss  the  variations  of  these  quantities  in  units 
of  the  last  decimal  place  of  the  logarithms. 

These  formula)  will  generally   be  applied  for  the  correction  of 


VAIUATIOX    OK   TUT)    GKOCKNTItlC    DISTANi^KH. 


ar 


a|)l)ro\imate  olcineuts  liy  iiicaiis  of  M'vcral  observed  places,  wliidi 
limy  hv  either  siiifxle  ohservatioiis  or  nonual  places,  eacli  derived  from 
K'Vt  ral  ohservations,  and  the  two  phu cs  selected  for  the  eoiupiitation 
(if  the  eh'inents  from  J  and  J''  should  not  only  l)e  the  most  accurate 
j)()ssil)le,  l)iit  they  shonld  also  he  such  that  the  resulting  elements  are 
not  too  much  all'eeted  by  small  errors  in  these  {geocentric  |)laces. 
Tliey  shotdd  moreover  be  as  distant  from  eacli  other  as  possible,  the 
other  c()nsiderations  not  being  overlooked.  When  the  three  systems 
of  elements  have  been  computed,  each  of  the  remaining  oltscrved 
places  will  furnish  two  equations  of  condition,  according  to  eipiations 
(10),  for  the  determination  of  tlie  corrections  to  be  applied  to  the 
nssumed  values  of  the  geocentric  distances;  and,  since  the  nund)er 
of  equations  will  thus  exceed  the  number  of  unknown  (piantities, 
the  entire  group  must  be  c(}nd)ined  according  to  the  method  of  least 
siiiiares.  Thus,  we  multiply  each  e<piation  by  the  eoellicieiit  of  aJ 
ill  that  equation,  taken  with  its  proper  algebraic  sign,  and  tlie  sum 
of  all  the  equations  thus  formed  gives  one  of  the  final  e(|uations 
required.  Then  we  multiply  eac^h  equation  by  the  coelficient  of  aJ" 
in  that  ccpiation,  taken  also  with  its  proper  algebraic  sign,  and  the 
sum  of  all  these  gives  the  second  equation  refjuired.  From  these 
two  final  equations,  by  elimination,  the  most  probable  values  of  aJ 
and  aJ"  will  be  obtained;  and  a  system  of  elements  computed  with 
tlio  distances  thus  corrected  will  exactly  represent  the  two  funda- 
nioiital  places  selected,  while  the  sum  of  the  squares  of  the  residuals 
for  the  other  places  will  be  a  minimum.  The  observations  are  thus 
supposed  to  be  equally  good ;  but  if  certain  observed  places  are 
entitled  to  greater  infiuence  than  the  others,  the  relative  precision 
of  these  places  must  be  taken  into  account  in  the  combination  of  the 
equations  of  condition,  the  process  for  whicih  will  be  fully  explained 
in  tlio  next  cha])ter. 

When  a  number  of  observed  places  arc  to  be  used  for  the  correction 
of  the  approximate  elements  of  the  orbit  of  a  planet  or  comet,  it  will 
be  most  convenient  to  adopt  the  equator  as  the  fundamental  plane. 
In  this  case  the  heliocentric  places  will  be  computed  from  the  assumed 
values  of  J  and  J",  and  the  corresponding  geoce'.:trie  right  ascensions 
and  declinations  by  means  of  the  formula}  (106)3  '"^'^^^  (l^');}  >  '^"^^  ^'^^ 
position  of  the  plane  of  the  orb! I:  as  determined  from  these  by  means 
of  the  equations  (76),  will  be  referred  to  the  equator  as  the  funda- 
mental plane.  The  formation  of  the  equations  of  condition  for  the 
corrections  aJ  and  aJ"  to  be  applied  to  the  assumed  values  of  the 
distances  will  then  be  effected  precisely  a.s  in  the  case  of  }.  and  ,-1,  the 


018 


TIIROUKTICAI-    ASTHONOMY 


lU'cossarv  clmnjijert  hv\u^  iimdc  in  tho  iiotiition.  In  a  similar  manner, 
tli(!  calculation  may  be  elU'ctcd  for  any  otiu'i*  I'undanicntal  phuu;  wliicli 
may  l»c  a(l<)i)tc(l. 

It  rilioiild  !)('  observed,  furtlicr,  that  when  the  ecliptic  is  taken  as 
the  f'nndamental  plane,  the  j;eocentrie  latitndes  shonid  he  convctcfj 
l»y  means  of  the  e(ination  ((J)^,  in  order  that  the  latitudes  of  the  sun 
shall  vanish,  otherwise,  for  strict  acenraey,  the  heliocentric  places 
must  be  deternuned  from  J  and  J"  in  accordance  with  the  eipiatioiis 
(89),. 

lOG.  The  partial  differential  cocHicients  of  the  two  spherical  co- 
ordinates with  respect  to  J  and  J"  may  be  computed  directly  by 
means  of  dill'ercntial  formuhe;  but,  excc[>t  for  special  cases,  tliu 
numerical  calculation  is  less  expeditious  than  in  the  case  of  tiie  indi- 
rect method,  while  the  liability  of  error  is  much  greater.  If  wc 
adopt  the  [)lane  of  the  orbit  as  determined  by  the  approximate  values 
of  J  and  J"  as  the  fundamental  plane,  and  introduce  ;f  as  one  of  tlio 
elements  of  the  orbit,  as  in  the  ecpiations  (72)^,  the  variation  of  tlio 
geocentric  longitude  9  measured  in  this  plane,  neglecting  terms  of  the 
second  order,  depends  on  only  four  elements;  and  in  this  ease  tlio 
differential  formulie  may  be  applied  with  fiieility.  Thus,  if  we  ex- 
press r  and  v  in  terms  of  the  elements  <f,  M^,  and  /i,  we  shall  have 


and 


01 


dr    dip    .     dr 
dip  '  dJ  "^  dM. 


dv 


djr 
dJ 

dv dv    d(p 

dJ~d^"dJ~^  "dM^ 


d  (v  ■-)-  x)       d/    ,    dv    d<p 


dJ 


dJ    ^  ()!> 


dMg    .    dr    dn 
dJ   "^  dji  '  dJ' 

dMg       dv    dfjL 
dJ '  '^  (Ifx  ■  dJ' 


dv 


dJ  "^  di\L 


dMn    .    dv     dfi 
dJ   '^'dJl'dJ' 


In  like  manner,  wc  have 


dr"      dr" 
dJ        dtp 

d<p        dr" 
■  dJ  "^  dM, 

dM,       dr" 
rfJ    ^  d/x 

dn 

'dJ' 

dd             d<p 

dip        dv" 
dJ  +  d3f. 

dM,       dv" 
dJ     '    d,i 

d/J.        dx 
■  dJ  "^  dJ 

,,  ,  ^   dr    d(v+/)    dr"        .   d{v" -\- x) 

As  soon  as  the  values  of    ^t'  — j  . — >    tt'  and  — — rv — -  are 

rtJ         dJ         dJ  «J 

known,  the  equations  necessary  for  finding  the  differential  coefficients 
of  the  elements  ;f,  ^,  Ji^,,  and  fi  with  respect  to  J  are  thus  provided. 
In  the  case  under  consideration,  when  an  increment  is  assigned  to  J, 


VAIUATIO.V   OF   TWO   OEOCENTItlC    DISTANf'KS. 


;J19 


tlio  value  of  J"  rctiminiiij;  uncliiuij^cil,  /•"  and  v"  f  X  '"'*^  •'•'^  cliaiigi'*!, 
and  iioiiuc 


dr" 


=  0, 


(/J 


=  0. 


To  tiiid   .  .  and        ,  . — ,  Irom  tiie  equations 

J  cos  ij  cos  0  =  X  -\-  X, 
J  cos  yj  sin  0  :=y  -\-  Y, 

in  wliicli  )y  is  the  goooentrlc  latitude  in  reference  to  the  plane  of  the 

orbit  coiiipiited  from  J  and  J"  as  the  fundamental  plane,  autl  A',   Y 

the  gooeentric  co-ordinates  of  the  sun  referred  to  the  same  plane,  wo 

sot 

dx  =  cos  Tj  cos  0  dJ, 

rfy  =  C0S1J  sin  <?f/J, 

or,  substituting  for  dx  and  di/  their  values  given  by  (73)j, 

cos  yj  cos  0  dJ  --^  cos  /(  dr  —  r  sin  a  d  (v  -f-  x^> 
COS  rj  sin  0  dJ  =^  sin  it  (/*•  -}-  r  cos  u  d{v  -{■  x)' 

Eliminating,  successively,  d{o  +  x)  and  f?/-,  we  get 

dr 


dJ 


=:  cos  tj  COS  (fl  —  t<)> 


,-,-^~  =  -  cos  1J  SUl  (0  —  U). 

dJ  r 


(12) 


(13) 


Therefore,  we  shall  liave 

(h   ,     dv     d<p    ,     dv      dJ\f„    ,     dv      dfi       1  .    ,„        . 

dJ  +  W'<^^^d^fo"<u'^^rdJ=r''''''"'^-"^' 

dr     dv    ,     dr      dM„    ,     dr      dfi  .„         , 

-d'i  '  dJ  +  dM:  •    dJ   +    d;.   '  dJ  =  '"' ''  '''  ^'  -  ''^' 
dx  j_df_   d^,df_    dM,       dv'^    ^^Ji—o 
dJ  '^  d<p  'dJ'^  di%  '   r/J   "'"   (///   *  rfJ  ~    ' 
dr"     d<p^,dr'^    dM^        dr"_    dtJ^_r. 
dip  '  dA  '^  (IM,  '   rfJ   "^    dti.   "(U  ""    ' 

and  if  we  compute  the  numerical  values  of  the  differential  coefficients 

of  /•,  /•",  V,  and  v"  with  respect  to  the  elements  <p,  Mq,  and  /i,  these 

(Hjuatious  will  furnish,  by  elimination,  the  values  of  the  four  un- 

1  x-i-      dx   d<p    dM„        ,  dn 

known  quantities  3^7,  -rr.  -,-r.  and  ^-r- 
^  rtJ   rfj     rtJ  a  J 

In  precisely  the  same  manner  we  derive  the  following  equations 


320 


THEORETICAL   ASTRONOMY. 


for  the  ti'^terinination  of  the  partial  differenthl  coefficients  of  those 
elements  with  respect  to  J": — 


-1T--Z,+ 


dJ" 

dv 
'^  d^ 

dtp 

dr 

dr 
d<F 

d<p 
dJ" 

dy. 

(U"' 

^  df 

dtp 

dr 

dv' 

dtp 

dip 

dJ" 

,r  + 


dv_ 

dMo 

dr 

dM: 


dv"_ 


+ 


dr" 
dM' 


dM^  ,    dv 

dr  +  d;i 

dM„  dr 

~dr  "^  dfl 
dM„       ihr_ 

dr  +  d;i 

dJf„  dr" 

dr  +  d/^ 


0, 
0, 


d'JL 

\ir 

dix 

dr 

dii 

dr 

'     ==  cos  ij"  cos  (<?"  ■ 


cos  >?"  sin  (0" 


(14) 
n'). 


u'  I 


Since  the  geocentric  latitude  tj  is  affected  chiefly  by  a  change  of  the 
position  of  the  plane  of  the  orbit,  while  the  variation  of  the  longitude 
C  is  independent  of  SI  and  l  when  the  squares  and  products  of  the 
variations  of  the  elements  are  neglected,  if  we  determine  the  elements 
which  exactly  represent  the  places  to  which  J  and  J"  belong,  as  well 
as  tiie  longitudes  for  two  additional  places,  or,  if  we  determine  those 
which  satisfy  the  two  fundamental  places  and  the  longitudes  for  any 
number  of  additional  observed  places,  so  that  the  sum  of  the  squares 
of  their  residuals  shall  be  a  minimum,  the  results  thus  obtained  will 
very  nearly  sati  .y  che  several  latitudes. 

Let  0'  denote  iiie  geocentric  longitude  of  the  body,  referred  to  the 
plane  of  the  or')it  computed  from  J  and  J"  as  the  fundamental  plane, 
for  the  date  r'  of  any  one  of  the  observed  places  to  be  used  ibr  cor- 
recting these  assumed  distances.  Then,  to  find  the  partial  differential 
coefficients  oi'  6'  with  res?)ect  to  J  and  J",  we  have 


COSIJ 


t/J 


,dO'      dy     ,  ,d(r      dtp 

'd^'dS--^''''''^'cW'dS 

,  do'      dii 


r-r.  COS  r,   -r-'        ■-  -f  COS  f)   j-  •       ,  r  +  ^OS  J)     ,  ,J^ 


,  do'     dM, 


dM^     dJ 


+  cos  ^ 


COS  5? 


,  do' 
dr 


dii      dJ 

,  do' 

COS  f)     .—  ■ 

dtp 
,dO'    _d:i^ 
dfi '  dJ" 


(15) 


,dO'     dy     ,  ,dO'     dtp     ,  ,  do'     dM, 

'''  "^  dx  •  dJ"-  + '''  ">  d^  •  dr  -  ^-  ^«^  '^  dii: '  "d^ 


+  cos  Tj 


and  by  means  of  the  results  thus  derived,  we  form  the  equation 


,dO^ 


cos  rj'  AO'  =  COS  7j''Jj\aJ-\-  COS  r/   V  .„  A  J". 


,  do' 

'  dr 


(16) 


A  fourth  observed  place  will  furnish,  in  the  same  manner,   he  addi- 
tional equation  required  for  finding  aJ  and  aJ".     If  more  than  two 


VARIATIOX   OF   TWO   GEOCENTRIC   DISTANCES. 


321 


observations  arc  used  in  addition  to  the  fundamental  plaee.s  on  .'liirJ' 
tlie  assumed  elements  as  derived  from  J  and  J"  are  based,  the  several 
longitudes  will  furnish  each  an  equation  of  condition,  and  the  most 
probi'blc  values  of  aJ  and  aJ"  will  be  obtained  by  eombininji;  the 
entire  group  of  equations  of  condition  according  to  the  method  of 
least  squares. 

107.  In  the  actual  application  of  these  foimuhe  to  the  correction 
of  the  approximate  elements,  after  all  the  preliminary  corrections 
have  been  api)lied  to  the  data,  we  select  the  proper  observed  places 
for  determining  the  elements  from  the  corresponding  assumed  dis- 
tances J  and  J",  according  to  the  conditions  which  have  alr(!ady  been 
stated,  and  from  these  we  derive  the  six  elements  of  the  orbit.  Since 
tlie  data  furnisiied  directly  by  observation  are  the  right  ascensions 
and  the  declinations  of  the  body,  the  elements  will  be  derived  in 
reference  to  the  equator  a',,  the  plane  to  which  the  inclination  and  the 
longitude  of  the  ascending  node  belong.  These  elements  will  exactly 
represent  the  two  fundamental  places,  and,  if  the  assumed  distances 
J  and  J"  are  not  much  in  error,  they  will  also  very  nearly  satisfy 
the  remaining  places. 

We  now  adopt  as  the  fundamental  plane  the  plane  of  thu  ap{)roxi- 
mate  orbit  thus  dotevmincd,  and  by  means  o;'  the  equations  (83).^  and 
(85).,,  or  by  means  of  (87)2,  writing  a,  (?,  Q',  and  /'  in  place  of  /,  ^9, 
SI,  and  /,  ''C'spectivcly,  we  compute  the  values  of  0,  iy,  anil  y  for  the 
dates  of  the  several  places  to  be  employed.  Then  the  re^jiduals  for 
each  of  the  observed  places  are  found  from  the  formultc 


cosi;  A6'  :r=  sin  y  A'5  -|-  cos^  cos '5  Aa, 
at;  =:  cos  >*  A'5  —  sin  ;*  cos  '5  Ao, 


(17) 


the  values  of  Aa  and  Ar?  for  each  place  being  found  l>y  subtracting 
from  the  obser\ed  right  ascension  and  declination,  respectivel} ,  the 
rijrht  ascension  and  declinntio!i  eom})uted  by  means  of  the  elements 
derived  from  J  and  *'.  The  values  of  d,  3j,  and  y  being  required 
only  for  finding  c^;^^  A^,  A;y,  and  the  differential  coelHcients  of  6  and 
ij,  with  respect  to  the  elements  0''  the  orbit,  need  not  be  determined 
with  great  accuracy. 

Next,  we  compute  j-:  and  TT      ^^O"'  equations  (12),  and  from 

/ip\    ii  1  £>  fl>'    dr"    dv    dv"    dr     r,        ,  ,.      1  •  . 

(lb).,  the  values  of      -,  ^— ,   — ,  -j-,  -— ,  cvc,  by  means  ot  which, 
dtp    d<p     d<f    dip    dM^ 

n,sing  the  value  of  u  in  reference  to  the  equator,  we  form  the  efpia- 

tioiis  (13).     The  accent  is  added  to  y  to  indicate  that  it  refers  to  the 

ai 


.322 


niEORETICAT.    ASTRO.VOJIY. 


equator  as  tlio  piano  for  defining  tlio  olenioiits.  Tliii^  we  obtain  four 
e(}uations,  from  wliieli,  hy  elimination,  the  valnes  of  tiic  (litferentiiil 
eoeffieients  of  y'^  tf,  J/,,,  and  //  with  respect  to  J  may  be  obtained. 
In  the  ninnerieal  .solution,  by  subtracting  the  third  eipiatiou  from 

the  first,  the  unknown  quantity  -.7  is  immediately  eliminated,  so  that 

we   iiave  three  equations  to  find  the  three  unknown  quantities  -— , 

-  r-p  and  y-7     These  having  been  found,  -7^.  may  be  obtained  from 

the  first  or  frjm  the  third  equation. 

In  the  .same  manner  we  form  the  equations  (14),  and  thence  derive 

Then,  by  means  of  the  for- 


the  values  of  ^„  ^„  ^,  and  ^_,„ 


mulai  ("6)2,  (78)^,  and  (79)2,  we  compute  for  the  date  of  each  pi;  1'" 
to  be  em[)loycd  in  <!orrccting  the  assumed  distances  the  vali';  ;  il 

cos;j'  -j-,,  coar/  7-7,  etc.,  and  hence  from  (15)  the  values  of  cos 35'-,- 

"X     do'    "V  .  .  .     «-J 

and  cos  r/  --— ,•     The  results  tluis  obtained,  together  with  the  residuals 

com])uted  by  mcann  of  the  equations  (17),  enable  us  to  form,  accord- 
ing to  (IG),  the  equations  of  condition  for  finding  the  values  of  the 
corrections  aJ  and  aJ".  The  .solution  of  all  the  equations  thus 
formed,  according  to  the  method  of  least  .'squares,  will  give  the  most 
probable  values  of  the.sc  quantities,  and  the  system  of  elements  wliii'h 
corresponds  to  the  distances  thus  cori'ccted  will  very  nearly  .satisfy 
the  entire  series  of  observations.  Since  the  values  of  cos  jj' A^'  are 
expressed  in  .seconds  of  arc,  the  resulting  values  of  aJ  and  aJ"  will 
also  be  expressed  in  .seconds  of  arc  in  a  circle  whose  radius  is  equal 
to  the  mean  distance  of  the  earth  from  the  sun.  To  express  them  in 
])arts  of  the  unit  of  space,  we  must  divide  their  values  in  seconds  01' 
arc  by  2()()2»J  1.8. 

The  corrections  to  be  applied  to  the  elements  computed  from  J  ami 
J",  in  order  to  .satisfy  the  corrected  values  J  +  aJ  and  J"  -j-  aj", 
may  l)e  computed  by  means  of  the  partial  diiferential  coefticieuts 
already  derived.     Thus,  in  the  case  of  ^',  wc  have 


i 


^^■=;S^^+^:>^^". 


from  which  to  find  a;^'  ;  and  in  a  similar  manner  A^,  aJ/(„  and  sn 

may  l)e  obtained.     It,  from  tlie  values  01  ~~"/T~  "  fnd rj„ — 1 

we  compute 


VARIATION   OF   TWO   GP:OC'EXTRIC   DISTANCES. 


323 


i.s  equal 


J  ami 


d(v-^x') 


^■/, 


and  apply  tli':sc  corrootions  to  the  values  of  v  aiul  v"  fouiul  tVoin  J 
and  J",  we  obtain  the  true  anoniidie.s  corresponding  to  the  distances 
J  +  A  J  and  J'"'  -\-  aJ".  The  corrections  to  be  applied  to  the  values 
of  /•  and  /•"  derived  from  J  and  J"  are  given  by 


A}' 


dr 
dJ 


aJ, 


ov 


If  aJ  and  aJ"  are  expressed  in  seconds  of  arc,  the  corresponding 
values  of  A/"  and  Ar"  must  be  divided  by  2002()4.8.  The  corrected 
results  thus  obtained  should  agree  with  the  values  of  r  and  /•"  com- 
])uted  <lirectly  from  the  corrected  values  of  i',  v",  j),  and  c  by  means 
of  the  polar  equation  of  tiie  conic  section.     Finally,  we  have 

dz  =  sin  ij  dJ, 

and  similaily  for  dz" ;  and  the  last  of  equations  (73)^  gives 


r  sin  ?<  Ai'  —  r  cos  it  sin  i'  A  $J '         ---  sin  ij  A  J, 
r"  sin  u"  Ay"  —  r"  cos  u"  sin  i'  ^Sl' =-  s\n  r/'  a  J", 


(18) 


from  which  to  find  a/'  and  aJ^',  u  and  a"  being  the  arguments  of 
tlie  latitude  in  reference  to  the  equator.     AVe  have  also,  according  to 

aui  =  a/  —  cos  ('  aQ', 


A-' 


A;f'  +  2sin^.Vt'AS^', 


i"oin  which  to  find  the  cori'cctions  to  be  applied  to  co'  and  z'.  The 
I  M  iiionts  which  refer  to  the  equator  may  thei  be  converted  into  those 
lor  the  ecliptic  by  ■  lOans  of  the  formuhc  which  may  be  derived  from 
{W^)^  by  intcrchang'-  '••  Q,  and  P'  and  180°  —  /'  and  /. 

The  iinal  residuals  o+*  the  longitiidcs  ivwy  be  obtained  by  substi- 
tuting the  adopted  vabies  of  aJ  and  aJ"  in  the  several  equations  of 
condition,  or,  which  affords  a  complete  proof  of  the  accuracy  of  the 
entire  calculation,  by  direct  calculation  from  the  corrected  elements ; 
and  the  determination  of  the  remaining  errors  in  the  values  of  iy  will 
sliow  how  nearly  the  position  of  the  plane  of  the  orbit  corresponding 
ti'  the  corrected  distances  satisfies  the  intermediate  latitudes. 

Instead  of  <p,  il/p,  and  /i,  we  may  introduce  any  other  elements 
which  determine  the  form  and  magnitude  of  the  orbit,  the  necessary 


324 


THEORETICAL   ASTRONOMY. 


changes  being  made  in  the  /brnuila).  Thus,  if  we  use  the  elements 
T,  (J,  and  e,  these  must  he  written  in  pbice  of  31^,  ft,  and  <f,  respect- 
ively, in  the  equations  (13),  (14),  and  (15),  and  the  partial  differential 
coeillcients  of  /•,  r",  v,  and  f"  with  respect  to  tlicse  elements  must  be 
coujputed  by  means  of  the  various  diiferential  formula)  which  have 
already  been  investigated.  Further,  in  all  these  cases,  the  homo- 
geneity of  the  formula)  must  be  carefully  attended  to. 

108.  The  approximate  elements  of  the  orbit  of  a  heavenly  body 
may  also  be  corrected  by  varying  the  elements  which  fix  the  position 
of  the  p]  >  H'  of  the  orbit.  Thus,  if  the  observed  longitude  and  lati- 
tude and  J  ues  of  ^  and  i  are  given,  the  three  equations  (91), 
will  contain  oj.  three  unknown  quantities,  namely,  J,  r,  and  u,  and 
the  values  of  these  may  be  found  by  elimination.  When  the  observed 
latitude  ,9  is  corrected  by  means  of  the  formula  (6)^,  the  latitudes  of 
the  sun  disapjiear  from  these  equations,  and  if  we  nmltiply  the  first 
by  sin  (O —  $^)sin/9,  the  second  (using  only  the  upper  sign)  by 
—  cos  ( O  —  SI)  sin  ,3,  and  the  tiiird  by  —  sin  {?.  —  O)  cos /5,  and  add 
the  products,  we  get 

sin /9  sin  (O  —  SI) 


tan  u  = T—. — 5-      .  _ 

cos  t  sm.j  cos  (O 

from  which  ic  may  be  found. 


SI)  —  sin  I  cos  ;5  sin  (A  —  ©  )' 


(19) 


If  we  multiply  the  second  of  those 
equations  by  siu;?,  and  the  third  by  —  cos /9  sin  (A  —  SI),  and  add  tiie 
products,  wc  find 

i?sin(0  —  SI) 


r  = 


sin  ^l  (sin  i  cot  /J  sin  (A  —  SI)  —  cos  i) 


(20) 


The  expression  for  r  in  terms  of  the  known  quantities  may  also  be 
found  by  combining  the  firs*,  and  second,  or  by  combining  the  first 
and  third,  of  equations  (91),.     If  we  put 


n  C09N-- 
?i  siniV- 


:  sin  (5  cos  (O 
cos  /?  sin  (A  — 


-Si), 

o). 


the  formula  for  m  becomes 
tan  T(  = 


cosiV 


cos  (iV-f  i) 


tan  (O  — Si). 


(21) 


The  last  of  equations  (91),  shows  that  sin  it  and  sin/9  must  have  the 
same  sign,  and  thus  the  quadrant  in  which  w  must  be  taken  is  deter- 
mined.    Putting,  also, 

7)1  cos  M  ^^  sin  u, 
-  ?Hsiu3f  =siuttcot/3sin(A  —  ft), 


VARIATION  OF  THE   NODE   AND   INCLINATION. 


n2o 


we  have 


r  = 


cosil/       Rs\i\(Q  --  9,) 


cos  ( i¥  +  i) 


sin  w 


(22) 


AVhen  any  other  phmc  is  takoa  as  the  f'unclamental  piano,  the 
latitude  of  the  sun  (which  will  then  refer  to  this  plane)  will  be  re- 
tained in  the  equations  (91)i  and  in  the  resulting  expressions  for  (t 
and  r. 

Tiie  value  of  u  may  also  be  obtained  by  first  computing  v  and  i// 
by  means  of  the  equ.-i^'  ns  (42)3,  and  then,  if  z  denotes  the  angle  at 
the  planet  or  comet  between  the  earth  and  sun,  the  values  of  u  and 
z,  as  may  be  readily  seen,  will  be  determined  by  means  of  the  rela- 
tions of  the  parts  of  a  spherical  triangle  of  Avhich  the  sides  are 
180°  —  (2  +  ij/),  180°  +  O  —  «,  and  u,  the  angle  opposite  to  the 
side  n  being  that  which  we  designate  by  w,  and  the  side  180°  +  O  —  S2 
being  included  by  this  and  the  inclination  i.  Let  *S'=  180°  —  (z  +  \|/), 
and,  according  to  Napier's  analogies,  this  spherical  triangle  gives 


w o  I      \       cos  i  (i  —  w) 

tan  i  (5  +  «)  = :-.-—.  — r 

'  cos  i  {%  +  w) 


coti($^  —  O), 


wc        ^       sinA(i  — w)      .,.^        -v 
tan  I  {S -  .)  =  --r^^^p^  cotUa-Q), 

from  which  S  and  u  are  readily  found.     Then  we  have 

g  =  180°  —  +  —  6; 

It  sin  ■4' 


(23) 


(24) 


r  = 


sm2 


to  find  r. 

If  we  assume  approximate  values  of  Q  and  /,  as  given  by  a  system 
of  elements  already  known,  the  equations  here  given  enable  us  to  find 
r,  n,  r",  and  «"  from  k,  /9  and  /",  ^9",  corresponding  to  the  dates  t 
and  t"  of  the  fundamental  places  selected,  and  from  these  results  for 
two  radii-vectores  and  arguments  of  the  latitude,  the  remaining 
elonients  may  be  derived.  From  these  the  gfocentric  place  of  the 
body  may  be  found  for  the  date  t'  of  any  intermediate  or  additional 
observed  place,  and  the  difference  between  the  computed  and  the 
observed  place  will  indicate  the  degree  of  precision  of  the  assumed 
values  of  SI  and  i.  Then  we  assign  to  Q  the  increment  oSi,  i 
remaining  unchanged,  and  compute  a  second  system  of  elements,  and 
from  these  the  geocentric  place  for  the  time  t'.  We  also  compute  a 
third  system  from  SI  and  /  +  di,  and  by  a  process  entirely  analogous 
to  that  already  indicated  in  the  case  of  the  variation  of  two  geocentric 


>^u 


THEOKKTICAL   ASTRONO.A/Y. 


distances,  wo  obtain  the  numerical  values  of  the  difrerential  eooffi- 
cicuts  of  /'  and  ^3'  with  respect  to  Ji5  and  /.     Thus  the  equations 


cos  /S'  aA'  =  cos  [i'  - ,  _   A  JJ  +  cos  /5'  -y  6,1, 


(25) 


for  finding  the  corrections  aS^  and  At  to  be  ajiplicd  to  the  assumed 
values  of  those  elements,  will  be  formed;  and  each  additional  obser- 
vation or  normal  place  will  furnish  two  equations  of  condition  for 
the  determination  of  those  corrections. 

If  the  observed  right  ascensions  and  declinations  are  used  directly 
instead  of  the  longitudes  and  latitudes,  the  elements  Q,  and  i  must 
be  referred  to  the  equator  as  the  fundamental  plane,  and  the  declina- 
tions of  the  sun  will  appear  in  the  formula)  for  u  and  r  o])tained  from 
the  equations  (91),,  thus  rendering  them  more  complex.  Their  deri- 
vation offers  no  difficulty,  being  similar  in  all  respects  to  that  of  the 
equations  (19)  and  (20),  and  since  they  will  be  rarely,  if  ever,  re- 
quired, it  is  not  necessary  to  give  the  process  here  in  detail.  In 
general,  the  equations  (23)  and  (24)  will  be  most  convenient  for 
finding  /•  and  u  from  the  geocentric  spherical  co-ordinates  and  the 
elements  Q,  aud  /,  since  %',  i//,  xo" ,  and  i^"  remain  unchanged  for  the 
tlircic  hypotheses. 

When  the  equator  is  taken  as  the  fundamental  plane,  4-  is  the 
distance  between  two  points  on  the  celestial  sphere  for  which  the 
geocentric  spherical  co-ordinates  are  vi,  D  and  a,  «?,  those  of  the  sun 
being  denoted  by  A  and  D.     Hence  we  shall  have 


sin  4  sin  B  =  cos  '5  sin  (a  —  A), 

sin  4  cos  £  =  cos  D  sin  d  —  sin  D  cos  li  cos  (a  —  A), 

cos  4-  :::=  sin  Z)  siu  (J  -|-  COS  D  cos  8  cos  (a  —  A), 


(26) 


from  which  to  find  t]/  and  B,  the  angle  opposite  to  the  side  90"^  —  d 
of  the  spherical  triangle  being  denoted  by  B.  Let  K  denote  the 
rigiit  ascension  of  the  ascending  node  on  the  equator  of  a  great  circle 
passing  througli  the  places  of  the  sun  and  comet  or  planet  for  the 
time  t,  and  let  w^  denote  its  inclination  to  the  equator;  then  we  shall 

have 

sin  «?,,  cos  (A  —  K)  =  cos  B, 

sin  u'o  siu  {A  —  K)  =  sin  B  sin  J),  (27) 

cos  it>.  =  sin  B  cos  D, 


from  which  to  find  w.  and  K.     In  a  similar  manner,  we  may  com- 


VARIATION  OF   THE   XOPE   AND   INCLINATION. 


327 


putc  the  values  of  ;/" — n,  ^,  and  /  from  the  holiocontric  sjihorical 
co-oriHiuite8  /,  ft  and  /",  b'\ 
From  the  ccjuatioud 

tan  ^iS,  +  u) 


tan  J  (.S',  —  u) 


cos  j  (/' — K'q) 

'cos.J(t'+  U'o) 
sin  I  (i'  —  ■!<'o'* 
sin  A  ( i'  +  icj 


cot\{Sl'~K), 
cot\{gi'-K), 


(28) 


the  accents  being  added  to  distinguish  the  elements  in  reference  to 
the  equator  from  those  with  respect  to  the  ecliptic,  the  values  of  <% 
and  u  (in  reference  to  the  equator)  may  be  found.  Let  ,\  denote  the 
angular  distance  between  the  place  of  the  sun  and  that  point  of  the 
equator  for  which  the  right  ascension  is  A",  and  the  ec^uatiou 


cot  Sq  =  COS  ?/'g  cot  (K  —  A) 


(29) 


gives  the  value  of ,%,  the  quadrant  in  which  it  is  situated  being  deter- 
mined by  the  condition  that  cos  s^  and  cos  (A' — ^i)  shall  have  the 


same  sign. 


Then  wc  have  *S'  =  aSq  —  s„,  and 

li  sin  -i^ 
sin  2 


(30) 


r  = 


from  which  to  find  r, 


109.  In  both  the  method  of  the  variation  of  two  geocentric  dis- 
tances and  that  of  the  variation  of  SI  and  /,  instead  of  using  the 
geocentric  spherical  co-ordinates  given  by  an  intermediate  observa- 
tion, in  forming  the  equations  for  the  corrections  to  be  applied  to  the 
assumed  quantities,  we  may  use  any  other  two  quantities  \vhi(;h  may 
be  readily  found  from  the  data  furnished  by  observation.  Thus,  if 
we  compute  /•'  and  u'  for  the  date  of  a  third  observation  directly 
from  each  of  the  three  systems  of  elements,  the  differences  between 
the  successive  results  will  furnish  the  numerical  values  of  th<!  partial 
differential  coefficients  of  r'  and  a'  with  respect  to  J  r.nd  J",  or  witii 
respect  to  Si  smd  i,  as  the  case  may  be.  Then,  computing  the  values 
of  /•'  and  u'  from  the  observed  geocentric  spherical  co-ordinates  by 
means  of  the  values  of  Q  a,nd  i  for  the  system  of  elements  to  be 
corrected,  the  diffijrences  between  the  results  thus  derived  and  those 
o'otuincd  directly  from  the  elements  enable  us  to  form  the  equations 


du'      .   ,    dti'       ,,,  , 

dr'      ,    ,     dr'       ,,,  , 


(31) 


328 


TIIEORETrcAL   A8TTIOXOMY. 


or  tlic  corrcHponcling  oxprcasioiis  in  the  case  of  the  variation  of  ^ 
and  /,  by  means  of  which  the  corrections  to  be  applied  to  the  as- 
sumed vaUies  will  be  determined.  In  the  numerictil  applic'ution  of 
these  equations,  Au'  being  expressed  in  seconds  of  arc,  a/-'  should  also 
l)e  expressed  in  seconds,  and  the  resulting  values  of  aJ  and  aJ"  will 
be  converted  into  those  expressed  in  parts  of  the  unit  of  space  by 
dividing  them  by  206264.8. 

When  only  three  observed  places  arc  to  be  use<l  for  correcting  an 
approximate  orbit,  from  the  values  of  r,  v',  r"  and  u,  u',  a"  obtained 
by  means  of  the  formula)  which  have  been  given,  we  may  find  p  and 

«  or the  latter  in  the  case  of  very  eccentric  orbits — from  the  first 

and  sec(»nd  places,  and  also  from  the  first  and  third  places.  If  those 
results  agree,  the  elements  do  not  require  any  correction;  but  if  a 
difference  is  found  to  exist,  by  computing  the  differences,  in  the  case 
of  each  of  these  two  elements,  for  three  hypotheses  in  regard  to  J 
and  J"  or  in  regai'd  to  Q,  and  /,  the  equations  may  be  formed  by 
means  of  which  the  corrections  to  be  applied  to  the  assumed  values 
of  the  two  geocentric  distances,  or  to  those  of  ft  and  /,  will  be 
obtained. 

110.  The  formulte  which  have  thus  far  been  given  for  the  correc- 
tion of  an  a])proximate  orbit  by  varying  the  geocentric  distances, 
depend  on  two  of  these  distances  when  no  assumption  is  made  in 
regard  to  the  form  of  the  orbit,  and  these  formulte  apply  with  equal 
facility  whether  three  or  more  than  three  observed  places  arc  used. 
But  when  a  series  of  places  can  be  made  available,  the  problem  may 
be  successfully  treated  in  a  manner  such  that  it  will  only  be  necessary 
to  vary  one  geocentric  distance.  Thus,  let  .r,  y,  z  be  the  rmnangular 
heliocentric  co-ordinates,  p.nd  r  the  radius-vector  of  the  body  at  the 
time  t,  and  let  A'^  Y.  Z  be  the  geocentric  co-ordinates  of  the  sun  at 
the  same  instant.  Let  the  geocentric  co-ordinates  of  the  body  be 
designated  by  Xq,  y^,  2„,  and  let  the  plane  of  the  equator  be  taken  as 
the  fundamental  plane,  the  positive  axis  of  x  being  directed  to  the 
vernal  equinox.  Further,  let  j>  denote  the  projection  of  the  radius- 
vector  of  the  body  on  the  plane  of  the  equator,  or  the  curtate  dis- 
tance with  respect  to  the  equator;  then  we  shall  have 


Tj  =  n  cos  a, 


yo  =  P  sm  a, 


2j  =  /)  tan  S. 


(32) 


If  we  represent  the  right  ascension  of  the  sun  by  A,  and  its  declina- 
tion by  D,  we  also  have 


VARIATION   OF   ONE  OKOCEXTRIC   DISTANCE.  329 

Xr=/2co8Z)cos.l,  F=/ecosZ)si!i^l,  Z=Iiii\nD.    (33) 

The  tundanieiital  equations  for  the  uiulisturbed  motion  of  tlie  planet 
or  comet,  neglecting  its  mass  in  comparison  with  that  of  the  sun,  are 


df  "^  V  ~  "'  dt'  "^  1^'  ~  "' 

but  since 

.r  =  a-o  —  X,  y  =  y^—Y, 

and,  neglecting  also  the  mass  of  the  earth, 


^a  -r  p3  —  "> 


R' 


d'Y      lc'Y_ 
{2     r    em"        '^» 


dt' 


B? 


d'z       k'z  __ 
df  +>  -"' 


3  =  Zo  —  Z, 


dt'  ^  Jte      ' 


these  becouie 


~di 


(34) 


Substituting  for  aT„,  ?/„,  and  ?;„  their  values  in  terms  of  a  and  d,  and 
putting 


we  get 


k-'p 

cos  a  -f-  ^  =  V, 

(36) 


d 


^+-fcOSa  +  ~0, 


Differentiating  the  equations  (32)  with  respect  to  t,  we  find 


Tt 


=  cos  a 


dp 
It 


dok 
/>sma^, 

diJa  .dp  da 

'-dt-'''"'di+f'''"'di' 

w-'''''"rt+'''''d-f 


(37) 


330 


TIIEOHETICAL   A8TH0NOMY. 


Ditlbrcnliatiiig  again  with  rospoct  to  t,  ami  .sub^itituting  iu  the  cfjuu- 
tions  (30)  the  vahics  thus  fouiul,  the  results  are 

ik'r  ,  d',>         <la:\  (    iPa         (I,,    (L\   .        .    ^       , 

(^+;;;;'-.:;;^)«in.+(.;;>2;;^.|)eos.-,,=o.     (38) 

(^  +  ;;;;')tan.  +  2.e..|.^  +  2.se..tan.;;;:  +  ..ec.|^  +  :.0. 

If  we  multiply  the  first  of  the.sc  equations  by  sin  a,  and  tlie  second 
by  —  cos  (X,  and  add  the  products,  we  obtain 


lit 


f  sini 


7j  cos  o  —  p 


(Pa 
(It^ 


da 
dt 


Now,  from  (35)  we  get 

?  sin  tt  —  rj  cos  a  =^  ^M  jm j)ii  COS  D  sin  (o  —  A), 

and  the  preceding  equation  becomes 


dp 

dt 


da 

dt 


(39) 


dp 


The  value  of  ~7  thus  found  is  independent  of  the  differential  co- 
efficients of  d  with  respect  to  t.     To  find  another  value  of  -A  using 

all  three  of  equations  (38),  we  multiply  the  first  of  tliese  equations 
by  sin  A  tan  o,  the  second  by  —  cos  A  tan  d,  and  the  third  by 
—  sin  (a  —  ^1).  Then,  adding  the  products,  since  ^  sin  A  =  rj  cos  A, 
the  result  is 

from  which  we  get 


dp 

di 


=  iP 


cot  (a  -A)^+  sec^  .3{  2  '£  +  cot  ^g-^ )  +  ^  coto      ^ 


^  i\  da  .  ,        ..  ,  «/t5 

cot  (o  —  A)- cot  S  sec'  8  — 

dt  dt 


(40) 


VARIATION'   OF  ONE   GEfK'KXTRIC   DTSTANTE. 


331 


W  hen  tlic  ecliptic  is  taken  as  the  fiindainental  plane,  the  last  term 
of  the  numerator  of"  the  .><ee()n(l  iiieii»l)er  of  this  eciuation  vanishen, 
and  the  e[)uation  may  be  written 


dp 
dt 


=  Cp, 


(41) 


tlic  coetricient  C  being  independent  of  (). 


dp 


111.  When  the  value  of  f>  is  given,  that  of  -j-  will  be  determined 

ill  terms  of  the  data  furnished  directly  by  observation  and  of  the 
dittorential  coefKcients  of  a  and  o  with  respect  to  t  from  ecpiation 
(3!)),  or  from  (40),  the  latter  being  i)referred  when  the  motion  of  the 

body  in  right  ascension  is  verj'  slow.     The  value  of  -^  having  been 

found,  we  may  compute  the  velocities  of  the  body  in  directions 
parallel  to  the  co-ordinate  axes.     Thus,  since 


a-o  =  .c  +  X, 
the  equations  (37)  give 


yo^y+  y, 


2o  =  2  +  ^, 


dx 
dt  ~ 

dp            .        da 
=  cos  a  ^-  —  p  sm  a  -r-  - 
dt                     dt 

dX 

dt' 

dy_ 

dt  ~ 

dp    ,               da 
=:,ma-+pCOSa-- 

dV 

di' 

dz  _ 
dt  " 

=  tan<J^+^sec^^|- 

dZ 

~di' 

(42) 


% 


bv  moans  of  which  -^,  -~,  and  ^  may  be  determined. 

dt   dt  dt       •' 

To  find  the  values  of  -,-,  -fr,  and  -77.  the  equations 

dt    dt  dt  ^ 


give,  by  differentiation, 


A'^^cosO, 
Y=  B»'m  O  cose, 
Z  ^=  E  sin  O  sin  e, 


dX  ^dR       „  .    ^  dQ 

'dt-''''^-dt-^''''^lu' 

dY       .    ^         dR  .    J.        .         dQ 

-rrr  =  Sm  ©  COS  £  —jr  +  R  COS  ©  COS  £  —jr-, 

dt  dt  dt 

dZ        .    ^    .      dR  ,    „        ^    .      dQ 
-,/  =  sm  ©  sm  e  -J.  +  if  cos  ©  sin  e  j-. 
dt  at  dt 


(43) 


i 


/ 


332  TIIEOUETICAL   ASTRONOMY. 

Now,  according  to  equation  (52),,  we  have 


dQ 
(It 


>l(l-c„0  (!  +  »«„) 


K' 


(41) 


?/)„  dciiotiu}^  tlie  mass  of  tlie  oartli,  and  (•^,  tlic  eccentricity  of  its  orbit. 
Tile  polar  equation  of  the  conic  .section  gives 


(It 


rh'.  sin  V    dv 
2>       '  (It' 


hct  r  denote  the  longitude  of  the  .sun's  perigee,  and  this  equation 
gives 

!„  (it 


(It 


1 


V\-e} 


If  we  neglect  the  .square  of  the  eccentricity  of  the  earth's  orbit,  we 
liave  simply 


do       ^"l/l  +  wio    dH 


(It 


The  values  of 


R' 


dt 


kVl  4-jnueosin(0  —  T). 


'3) 


a© 


dR 


-r-  and    -,     having  been  found  by  means  of  these 

.  .  dX  dY 

formula?,  the  equations  (43)  give  the  required  results  Ibr  ~,     -.  and 

dZ 

-jj,  and  hence,  by  means  of  (42),  we  obtain  the  velocities  of  the 

comet  or  planet  in  directions  parallel  to  the  co-ordinate  axes. 

112.  The  values  of  x,  y,  and  z  may  be  derived  by  means  of  the 

equations 

x=  d  cos '5 cos  a  —  X, 
y  :=  J  cos  d  sin  o  —  F, 

z  —  d  sin  d  —  Z, 

and  from  these,  in  connection  with  the  corresponding  velocities,  the 
elements  of  the  orbit  may  be  found.  The  equations  (-32),  give  im- 
mediately the  values  of  the  inclination,  the  semi-parameter,  and  the 
right  ascension  of  the  ascending  node  on  the  equator.  Then,  tlie 
position  of  the  plane  of  the  orbit  being  known,  we  may  compute  r 
and  u  directly  from  the  geocentric  right  ascension  and  declination  by 
means  of  the  equations  (28)  and  (30).  But  if  we  use  the  values  of 
the  heliocentric  co-ordinates  directly,  multiplying  the  first  of  equa- 
tions (93),  by  cos  SI,  and  the  second  by  sin  ft,  and  adding  the  pro- 
ducts, we  have 


VARIATION    OF   OXK   (JEOCKNTIIIC   DISTANX'E. 


333 


(47) 


»•  Hill  It  =--«  cosec  (, 

r  c'Ori  It  =  X  (Mw  SI  -\-  y  »*'»  SI, 

from  wlilcli  r  iiiul   it  may  l)o  found,  the  !ii'f;uinent  of  the  latitiuloJt 
Ixinu;  ri'ft'rrod  to  tlie  phuiu  ol'  .17/  a.s  tlie  I'uiuUuneutal   plane.     The 

e(iiiatiou 

r»  =  a;'  -f-  y' +  z' 


gives 

and,  since 
we  t^hall  have 


dr X    dj;      y    dy       z    dz 

dt       }•  '  dt       r  '  dt       r  '  dt ' 


(48: 


dr )''e  sin  v    dv 

di  "^      p        di' 


dv k  Vp 

dt  ~  ~r'    ' 


Vp    dr 


ecosv 


P 


(49) 


'--1. 
r 


from  which  to  find  e  and  v.     Then  the  distance  between  the  ytcvi- 
helion  and  the  ascending  notle  is  given  by 


w  =  ?t 


The  sonii-transverse  axis  is  obtained  from  p  and  e  by  means  of  the 

relation 

P 

1  — e^ 

Finally,  from  the  value  of  v  the  eccentric  anomaly  and  thence  the 
moan  anomaly  may  be  found,  and  the  latter  may  then  be  referred  to 
any  epoch  by  means  of  the  mean  motion  determined  from  n. 

In  the  case  of  very  eccentric  orbits,  the  perihelion  distance  will  be 
given  by 

P 

and  the  time  of  perihelion  passage  may  be  found  from  v  and  c  by 
means  of  Table  IX.  or  Table  X.,  as  already  illustrated. 

The  equation  (21)i  gives,  if  we  substitute  for  /  its  value  in  terms 
of  p,  denote  by  V  the  linear  velocity  of  tlie  planet  or  comet,  and  neg- 
lect the  mass, 

dr'' 

Let  \//|3  denote  the  angle  which  the  tangent  to  the  orbit  at  the  ex- 
tremity of  the  radius-vector  makes  with  the  prolongation  of  this 
radius-vector,  and  we  shall  have 


334 


THEORETICAL   ASTKON<      Y. 


rVcoS'^g- 


dr 

'lit 


dx    ,      dy    ,       dz 

""'dt-^y-dt-^'it' 


so  that  the  prccct^ng  equation  gives 

Hence  we  derive  +^'T  equations 

Vr  sin  ^^  =  kVp, 

,.  dx    ,      dii    , 


dz 


(50) 


from  which  Vr  and  i^q  may  be  found.     Then,  since 


wc  shall  have 


k' 
a 


W 


-V, 


(51) 


by  means  of  which  a  may  be  determined,  and  then  e  may  be  foiiml 
by  means  of  this  and  the  value  of  p. 
The  equations  (49)  and  (50)  give 


e  sin  (ii  —  w) 


and,  since 


-^rsm  4-0  cos  ,1.0, 

V 
e  cos  (ii  —  w)  =  -,-  r  sin'  ^^  ■ 


1, 


k' 


a 


these  arc  easily  transformed  into 

2ae  sin  (u  —  w)  =  (2a  —  r)  sin  2-^^,, 

2«c'.^  cos  (n  —  w)  =  —  ( 2a  —  r)  cos  2-i>(,  —  }•. 

If  we  multiply  the  first  of  these  equations  by  —  cos  (i  and  the  .second 
by  sin  w,  and  add  the  products;  tiien  multiply  the  first  by  siu«  and 
the  second  by  cos  u,  and  add,  M'ti  obtain 

2ae  sin  w  =  —  (2('-  —  r)  sin  {2'^g  +  n)  —  r  sin  u,  ,r.y-. 

2oecoswr^  -■(2a  —  ?•)  cos(24/„ -f  (t)  —  rcosw, 

These  equations  give  the  values  of  (n  and  e. 

113.  We  have  thus  derived  all  the  formula)  necessary  for  findins,' 
the  elements  of  the  orl>it  of  a  heavenly  body  from  one  geocentric 
distance,  provided  that  the  first  and  second  diffenintial  coefficients  of 
a  and  d  with  respect  to  the  time  are  accurately  known.     It  remains, 


VARIATIOX  OF   OXE  GEOCENTRIC  DISTAXCE. 


335 


thoroforc,  to  devise  the  means  by  whieli  these  differential  cdefiieients 
niiiy  be  determined  with  accuracy  from  the  data  furnished  by  obser- 
vation. The  approximate  elements  derived  from  three  or  from  a 
small  number  of  observations  will  enable  us  to  correct  the  entire 
series  of  observations  for  parallax  and  r')erration,  and  to  f  trm  the 
normal  places  which  shall  represent  the  series  of  ol)scrved  places. 
We  may  now  assume  tiuit  the  deviation  of  i.he  spherical  co-ordinates 
conii)utcd  by  means  of  the  approximate  elements  from  those  which 
woidd  be  obtained  if  t!>e  true  elements  were  used,  may  be  exactly 
represented  by  the  formula 

c.O^A  +  Bh  f  Ch\  (53) 

h  denoting  the  interval  between  the  time  at  which  the  deviation  is 
expressed  by  A  and  the  time  for  which  this  ditfercnee  is  c^d.  The 
f^iacrences  between  the  normal  places  and  those  computed  w'ith  the 
approximate  elements  to  be  corrected,  will  then  sulfice  to  form  ccpia- 
tions  of  condition  by  mf^ans  of  which  the  values  of  the  coellii'ients 
A^  B,  and  C  may  be  determined  The  epoch  for  vln'ch  h  ---^-  0  may 
be  chosen  arbitrarily,  but  it  will  j^cncrally  be  advantajj;cous  to  fix  it 
at  or  near  the  date  of  the  middle  observed  place.  If  three  observed 
places  are  given,  the  difference  between  the  observed  and  the  com- 
puted value  of  each  right  aseeni-ion  will  give  an  equation  of  condition, 
according  to  (53),  and  the  three  etpiations  thus  formed  will  furnish 
the  numerical  values  of  A,  B,  and  C.  These  having  been  deter- 
mined, the  c({uation  (53)  will  give  the  correction  to  be  a])plied  to  the 
computed  right  ascension  for  any  date  within  the  limits  of  the 
extreme  observations  of  the  series.  When  more  than  three  normal 
places  are  determined,  the  resulting  equations  of  condition  may  be 
reduced  by  the  method  of  least  squares  to  three  final  ccjuations,  from 
which,  by  elimination,  the  most  probable  values  of  A,  B,  and  C'will 
1)0  derivaxl.  In  like  manner,  the  corrections  t  >  be  ai)plicd  to  the 
coinpufed  latitudes  may  be  determined.  The.se  corrections  being 
npplicd,  the  ephcmeris  thus  obtained  may  bo  assumed  to  rcpr(>sent 
the  apparent  path  of  the  body  with  great  precision,  and  may  be  cm- 
ployed  as  an  auxiliary  in  determining  the  values  of  the  differential 
coefficients  of  a  and  d  with  respect  to  t, 

liOt  f(a)  denote  the  right  ascension  of  the  body  at  the  middle 
epoch  or  that  for  which  h  ---  0,  and  lct/(«  ±  nio)  denote  the  value  of 
a  I'c  any  other  date  separated  by  tlie  interval  tuo,  in  which  (o  is  the 
interval  between  the  successive  dates  of  the  ephemeris.  Then,  if  we 
put  n  successively  erpial  to  1,  2,  3,  &c.,  wc  shall  have 


336  TIIEOKETICAL  ASTR0N05IY. 

Function.  I.  Diff.  II.  Diff.  III.  Diff".         IV.  Difl'.  V.  Diff. 


/'"(«+«>) 


/(a-2-)P"~!"^/"(«-2«>) 

/(a  — w)    J 

f(C')  fir      II    -\ 

/(«  +  ")  V„TL  /("■+■"'  /'"(»+<») 

/(a  +  oo;) 

The  series  of  functions  and  diifercnees  may  be  extended  in  the  same 
manner  in  either  direction.  If  we  expand  f{a  +  no))  into  a  series, 
the  result  is 

i-r  I  \  I       ^^*  ■       1    '^"''*      2     2      11    '^^''*      3     3      1         1     ''**      «     4      I       f 


(/< 


dt' 


df 


or,  puttnig  tor  brevity  A  =  -j:  <o,  is  =  ^  ^-7  w,  &c., 

/(a  +  ?iw)  =  tt  +  An  +  5/1^  +  C/i"  +  Dn*  +  &c. 

If  we  now  put  n  successively  equal  to  — 4,  — 3,  — 2,  — 1,  — 0,  +1, 

&c.,  we  obtain  the  values  of /(a  —  Aio),f(a  —  3w), f{a  -f-  -Iw) 

iu  terms  of  A,  B,  C,  &c.  Then,  taking  the  successive  orders  of 
differences  and  symbolizing  them  as  indicated  above,  we  obtain  a 
series  of  equations  by  means  of  which  A,  B,  O,  &c.  will  be  deter- 
mined in  terms  of  the  successive  orders  of  diffei'enecs.  Finally,  re- 
placing A,  By  C,  &G.  by  the  quantities  which  they  represent,  and 
putting 

ij'"  («  -  »  +  y"  (a  +  >)  =/'"  (a ),  &c., 
we  obtain 


da  _  1 
dt        w 

dt' 
d*a 
dt* 
d'a 
'(if 


(/'(«)  -  J/"'(«)  +  .'o/'W  -  rhr^a)  +  &c.), 


1 


:.-  ir  («)  -  .1/''  («)  +  2I 0/""  («)  -  &c.), 


(54) 


dt 


£=7!r(r(«)-&c.). 


^)  =  -.\  (/""(«) -&c.). 


VARIATION  OF  ONE   GEOCENTRIC   DISTANCE. 


337 


by  means  of  which  the  successive  differential  coefficients  of  a  with 
respect  to  /  may  be  determined.  The  derivation  of  these  coefficients 
in  the  case  of  o  is  entirely  analogous  to  the  process  here  indicated  for 
a.  Since  the  successive  differences  will  be  expressed  in  seconds  of 
arc,  the  resulting  values  of  the  diff'erential  coefficients  of  a  and  u  with 
respect  to  t  will  also  be  expressed  in  seconds,  and  must  be  divided  by 
206264.8  in  order  to  express  them  abstractly. 

,TT  ■,  Til  1  n  ff"'  d'^a  (Id  (I'o  ,       , 

>V  e  may  adopt  directly  the  values  oi  -^.  -^i  -vr.  and  -.-j  determinea 

by  means  of  the  corrected  ephcmcris,  or,  if  the  observed  places  do 
not  include  a  verv  long  interval,  we  may  determine  only  the  values 

of  jj,  -,.^-,  &c.  by  means  of  the  ephemeris,  and  then  find  -,-  and  y- 

directly  from  the  normal  places  or  observations.  Thus,  let  a,  a',  a" 
be  three  observed  right  ascensions  corresponding  to  the  times  t,  V,  t", 
and  we  shall  have 


da 

da', 


(f-o+.A5.'a'-o 


df 
d'a' 


'+'^v'-n+i^ 


^a' 

'  df' 

d'a' 


ij 


d*a 


d*a'. 


(t''-n'+kim(-^''-n'-^,Y'j^(f~ty+&c., 


d^ 


di* 


which  give 


I  +  U.--0 


,sOPa'  _a 


df 

d^a' 
d^ 


tf—t 


de 
,rfV 


di* 


&c., 


(55) 


K^"-0^^ 


^(*"-oS. -&«• 


df 


da 


These  equations,  being  solved  numerically,  will  give  the  values  of   .- 
and  we  may  thus  by  triple  combinations  of  the  observed 


and 


df 


places,  using  always  the  same  middle  place,  form  equations  of  con- 
dition for  the  determination  of  the  most  probable  values  of  these 
differential  ecefficienta  by  the  solution  of  the  equations  according  to 
the  method  of  least  squares. 

In  a  similar  manner  the  values  of   r;  and  -,  -  may  be  derived. 


dt 


dt 


114.  In  applying  these  formula;  to  the  calculation  of  an  orbit, 
after  the  normal  places  have  been  derived,  an  ephemeris  should  be 
eonij)utod  at  intervals  of  four  or  eight  days,  arranging  it  so  that  one 
of  the  dates  shall  correspond  to  that  of  the  middle  observation  or 
normal  place.     This  ephemeris  should  be  computed  with  the  utmost 

22 


338 


THEORETICAL   ASTRONOMY. 


care,  since  it  is  to  be  employed  as  an  auxiliary  in  determining  quan- 
tities on  M-liich  depends  the  accuracy  of  the  final  results.  The  com- 
parison of  the  ephemeris  with  the  observed  places  will  furnish,  by 
means  of  equations  of  the  form 

A  +  Bh  +  Ch'  =  Aa', 
A'  +  B'h  +  J'h'  =  Ad', 

h  being  the  interval  between  the  middle  date  t'  and  that  of  the  ])lace 
used,  the  values  of  A,  B,  C,  A',  &c.;  and  the  con-ections  to  be 
applied  to  the  ephemeris  M'ill  be  determined  by 

A    +  Bntu    4-  Cii'"'^    =  Aa, 

•      A'  +  B'noj  -\-  C'li'io^  =  Ad. 

The  unit  of  h  may  be  ten  days,  or  any  other  convenient  interval, 

observing,  however,  that  mo  in  the  last  equations  must  be  expressed 

in  parts  of  the  same  unit.     With  the  ephemeris  thus  corrected,  Ave 

da  (Pa  dd  d-d  ,        ,  1   •      1       mi 

compute  the  values  oi  -j:'  Zui'  Ti'  ^       IF^  ^^  ^^^^^^^y  explamed.     iJiese 

differential  coefficients  should  be  determined  with  great  care,  since  it 
is  on  their  accuracy  that  the  subsequent  calculation  principally  de- 
pends.    We  compute,  also,  the  velocities  ~,  ~~,  and  -jr  ^7  means 

dQ  dR  ^^     '^^  ^' 

of  the  formulae  (43),  -j-  and  —  being  computed  from  (46).    The 

quantities  thus  far  derived  remain  unchanged  in  the  two  hypothesis 
with  regard  to  J. 

Then  we  assume  an  approximate  value  of  J,  and  compute 

/J  =  J  cos  5 ; 

and  by  means  of  the  ((juation  (40)  or  (39)  we  compute  the  value  of 
-^7-     It  will  be  observed  that  if  we  put  the  equation  (40)  in  the  form 


p 

the  coefficient  ^  remains  the  same  in  the  two  hypotheses.     The  three 

,  V  ^  (Jp 

equations  (38)  mav  be  so  combined  that  the  resulting  value  ot  -^ 

will  not  contain  -^--     This  trausformatio  :  is  easily  effected,  and  may 

be  advantageous  in  special  cases  for  which  the  value  of  --t—  is  very 

Civ 

uncertain. 

The  heliocentric  spherical  co-ordinates  will  be  obtained  from  the 


RELATION   BETWEEN   TWO   PLACES  IN  THE  ORBIT. 


339 


assumed  value  of  J  by  means  of  the  equations  (106)3,  ^^^  *^^^  ^^" 
tangular  co-ordinates  from 

X  =  r  cos  b  cos  I, 
y^=^r  cos  h  sin  /, 
3  =  ?'  sin  b. 

The  velocities  -jr,  -■-,  and  -rr  will  be  given  by  (4r*),  and  from  these 

and  the  co-ordinates  x,  y,  z  the  elements  of  the  orbit  will  be  riom- 
puted  by  means  of  the  equations  (32)i,  (47),  (49),  &c.  AVith  the 
elements  thus  derived  we  compute  the  geocentric  places  for  the  dates 
of  the  normals,  and  find  the  differences  between  computation  and 
observation.  Then  a  second  system  of  elements  is  computed  from 
J  +  0  J,  and  compared  with  the  observed  places.  Let  the  ditt'crcnce 
between  computation  and  observation  for  either  of  the  two  spherical 
co-ordinates  be  denoted  by  n  for  the  first  system  of  elements,  and  by 
n'  for  the  second  system.  The  final  correction  to  be  applied  to  J,  in 
oroer  that  the  observed  place  may  be  exactly  i-eprcsented,  will  be 
determined  by 

^(«'-»)  +  «-0.  (56) 

Each  observed  right  ascension  and  each  observed  declination  will 
thus  furnish  an  equation  of  condition  for  the  determination  of  aJ, 
observing  that  the  residuals  in  riglit  ascension  should  in  each  case  be 
multiplied  by  cos  rj.  Finally,  the  elements  which  correspond  to  the 
geocentric  distance  J  +  a  J  will  be  detcrpiined  either  directly  or  by 
intorpolation,  and  these  must  rej)resent  the  entire  series  of  obterved 
places. 


115.  The  equations  (02)3  enable  us  to  ^...d  two  radii-vectores  when 
the  ratio  of  the  corresponding  curtate  distances  is  known,  provided 
that  an  additional  equation  involving  r,  r",  K,  and  known  quantities 
is  given.  For  the  special  case  of  parabolic  motion,  this  additional 
efpiation  involves  only  the  interval  of  time,  the  two  radii-vectores, 
and  the  chord  joining  their  extremities.  The  corresponding  equation 
for  the  general  conic  section  involves  also  the  semi-transverse  axis 
of  the  orbit,  and  hence,  if  the  ratio  M  of  the  curtate  distances  is 
known,  this  equation  will,  in  connection  with  the  equations  (52)3, 
enable  us  to  find  the  value  of  r  and  r"  corresponding  to  a  given 
value  of  a.    To  derive  t!ns  expression,  let  us  resume  the  equations 


'MO  THEORETICAL   ASTRONOMY. 

f.==E"-E-2e  sin  i  (E"  -  E)  cos  -J  (E"  +  -E),       .„. 
r  +  r"  =-  2a  —  2ac  cos  ^  (£"  —  £)  cos  ^  (£"  +  £). 
For  the  chord  x  we  have 

x'  ==  (r  +  r")*  —  4rr"  cos'  i  («"  —  u), 
which,  by  meaii-  of  (58)4,  gives 

«'  =  (r  +  r'O' 

—  4a'^  (cos'  I  {E"—E)-2e  cos  i  {E"—E)  cos  ^  (J?'^+S)+e»  cos"  1  (!:''+ J?)) ; 

and,  substituting  for  r  +  f"  its  value  given  by  the  last  of  equations 
(57),  we  get 

x^  =  4a'  sin'  ■}  {E"  —  E){1—  e'  cos'  ^{E"  +  £)).  (58) 

Let  us  now  introduce  an  auxiliary  angle  h,  such  that 

cos  h  =  e  cos  ^  (j;"  +  E), 

the  condition  being  imposed  that  h  shall  be  less  than  180°,  and  put 

g  =  ^{E"-E)', 

then  the  equations  (57)  and  (58)  become 

t' 

—  =  2g  —  2  sin  g  cos  h, 

r  -|-  »•"  =  2a  (1  —  cos  g  cos  li), 
K  =  2a  sin  g  sin  h. 
Further,  let  us  put 

h  —  g  =  d,  h-{-g  =  e, 

and  the  last  two  of  equations  (59)  give 

r  -|-  r"  +  X  =  'la  sin"  Js, 
r  +  ?•"  —  X  =  4a  sin'  ^S. 

Introducing  d  and  e  into  the  first  of  equations  (59),  it  becomes 

—  =  (e  —  sin  e)  —  (5  —  sin  S). 
a2 


(59) 


(60) 


(61) 


The  formuljB  (60)  enable  us  to  determine  e  and  d  from  r  +  '>'",  ^j 
and  a,  and  then  the  time  t'  =^k  (t"  —  t)  may  be  determined  from 
(61).     Since,  according  to  (58)^, 

v'n'"co8^(n"  —  u)=a{cosg  —  cos  A)  =  2  sin  ^e  sin  ^5, 


■'tELATION   BETWEEN  TWO  PLACES   IN  THE  OEBIT. 


341 


and  since  sin  ^e  is  necessarily  positive,  it  appears  that  when  n"  —  u 
exceeds  180°,  the  vahxe  of  sin  J5  must  be  negative,  and  wlien 
n"  —  it  =  180°,  we  have  ^  =  0;  and  thus  the  quadrant  in  which 
d  must  be  taken  is  determined.  It  will  be  observed  that  the  value 
of  ie,  as  given  by  the  first  of  equations  (60),  may  be  either  in  tlie 
first  or  the  second  quadrant;  but,  in  the  actual  application  of  the 
formulfe,  the  ambiguity  is  easily  removed  by  means  of  tae  known 
circumstances  in  regard  to  the  motion  of  the  body  during  the  in- 
terval t"  —  t. 

In  the  application  of  the  equations  (52)3,  by  means  of  an  approxi- 
mate value  of  X  we  compute  d,  and  thence  r  md  r".  Then  we  com- 
pute e  and  o  corresponding  to  the  given  value  of  a,  and  from  (61) 
we  derive  the  value  of 

r— <  =  r- 
k 

If  this  agrees  with  the  observed  interval  t"  —  t,  the  assumed  value 
of  K  is  correct;  but  if  a  difference  exists,  b}'  varying  )t  we  may 
readily  find,  by  a  few  trials,  the  value  which  \s'll  oxactly  satisfy  the 
equations.  The  formulaj  (70)3  will  then  enable  us  to  determine  the 
curtate  distances  p  and  p",  and  from  these  and  the  observed  spherical 
co-ordinates  the  elements  of  the  orbit  may  be  found. 

As  soon  as  the  values  of  u  and  u"  have  been  computed,  since 
€  —  ^  ==  E" —  E,  we  have,  according  to  equation  (85)^, 


cos 


sin  J-  («"—  tt)  /— 77 


which  may  be  used  to  determine  (p  when  the  orbit  is  very  eccentric. 
To  find  p  and  q,  we  have 


p  =  a  cos'  <p, 


5  =  2a  sin' (45°  — ^y); 


and  the  value  of  to  may  be  found  by  means  of  the  equati'    o  (S?)^  or 
(88),. 

116.  The  process  here  indicated  will  be  applied  chiefly  in  the  de- 
termination of  the  orbits  of  comets,  and  generally  for  cases  in  which 
a  is  large.  In  such  cases  the  angles  e  and  d  will  be  small,  so  that 
the  slightest  errors  will  have  considerable  influence  in  vitiating  the 
value  of  t"  —  t  as  determined  by  equation  (61);  but  if  we  transform 

this  equation  so  as  to  eliminate  the  divisor  a^  in  th^  first  member,  the 
uncertainty  of  the  solution  may  be  overcome.    The  difference  e — sine 


342 


THEORETICAL   ASTRONOMY. 


may  be  expressed  l)y  a  series  which  converges  rapidly  when  e  is  small. 
Thus,  let  us  put 


e  —  sin  £  =  y  si'i'  2^. 


X  =  sm 


5-U 
4"» 


and  wc  have 

-rf  =  2  cosec  -hs  —  ~^y  cot  ^£, 

Therefore 

— -  =  4cosec  J,£. 
dx 

dy  _8—  6y  cos ^£  _  4  —  3i/(l  -  2x) 

dx             sin''  -^£                  2a;  (1  —  x) 

If  we  suppose  y  to  be  expanded  into  a  series  of  the  form 

y  =  a-\-i3x-\-  yx^  -\-  dx^  +  &c., 
we  get,  by  differentiation, 

^  =  /3  +  2/-^;  +  35a;'  +  &c., 

and  substituting  for  -—  the  value  already  obtained,  the  result  is 

2/?a;  +  (4y  —  2;3)  x*  +  (6<J  —  4^)  x^  +  &c.  =  4  —  3a  +  (6a  —  3,3)  a; 

4-  (6/3  —  3;-)  x»  +  (6r  —  3'5)  a;*  +  &c. 
Therefore  w'C  have 


4  — 3a  =  0, 
6/?-3r  =  4r-2/3, 


60  — 3/3=2,3, 

ey  —  2d=^  65  —  4^, 


from  which  we  get 


4^  _4.6.8 

'3.5'        ^ "~  3  . 5  .  7' 


,      4.6.8.10   „ 
^=  3.5.7.9'*'- 


Hence  we  obtain 

e_sine=:=.4sin»^£(l+|sin'-]£+|^sin'i£+|^sin«i£  +  &c.),(62) 

and,  in  like  manner, 

3-sin3=|sin'^<j(l+|sin»i5+g— |8in«j«5+|^^sin«i5+&c.),  (63) 

which,  for  brevity,  may  be  written 

e  —  sms  =  iQ  sin»  ^e,  ^q^-^ 

8  —  ain3  =  ^Q'siii^d, 


RELATION   BETWEEN   TWO  PLACES   IN   THE  OKBIT. 


343 


Combining  these  expressions  witii  (61),  and  substitnting  for  sin  ^s  and 
sin  |(J  their  values  given  by  the  equations  (60),  there  results 


6/  =  §  (r  +  /'  +  x)3  rp  Q'  (,.  +  r" -  x)K 


(65) 


the  upper  sign  being  used  when  the  lieliocentric  motion  of  tlie  body 
is  less  than  180°,  and  the  lower  sign  when  it  is  greater  than  180°. 
The  coethcients  Q  and  Q'  rej)rescnt,  respectively,  the  series  of  terms 
enclosed  in  tlie  parentheses  in  the  second  members  of  the  ecpiations 
(62)  and  (6;»),  and  it  is  evident  tiiat  their  values  may  be  tabulated 
with  the  argument  s  or  J,  as  the  case  may  be.  It  will  be  observed, 
however,  that  the  first  two  terms  of  the  value  of  Q  are  identical  with 

the  first  two  terms  of  the  expansion  of  (cos^s)"''^  into  a  series  of 
ascending  powers  of  sin  ^e,  while  the  difference  is  very  small  between 
the  coefficients  of  the  third  terms.     Thus,  we  liavo 


(cos  is)-  V  =  (1  -  sin»  J £)-§  :^  1  +  i  sin'  \s  + 


6.  11 
5.  10 


sm 


♦  1, 


+ 


and  if  we  put 


we  shall  have 


11.16  .  ,,     ,    „ 


Q^ 


B. 


(cos  .10 


^„  =  1  +  ^9,sin«].  +  5\5ij^sin«i.  +  &C 
In  a  similar  manner,  if  we  put 

we  find 


b: 


(eosJtJ)3" 
^'  ==  1  +  T?s  sinM-^  +  2^  sin'  \5  +  &c. 


(66) 


(67) 


(68) 


(69) 


Table  XV.  gives  the  values  of  /?„  or  B^'  corresponding  to  e  or  3  from 
0°  to  60°. 
For  the  case  of  parabolic  motion  we  liave 


<2  =  i, 


Q'  =  \, 


and  the  equation  (65)  becomes  identical  with  (56)3. 

In  the  application  of  these  formula?,  we  first  compute  e  and  d  by 
means  of  the  equations  (60),  and  then,  having  found  B^  and  B^'  by 
means  of  Table  XV.,  we  compute  the  values  of  Q  and  Q'  from  (66) 
and  (68).  Finally,  the  time  t'=  k{t"—t)  will  be  obtained  from  (65), 
and  the  difference  between  this  result  and  the  observed  interval  will 


344 


THEORETICAL   ASTRONOMY. 


indiciitc  whether  the  ussiuned  vahic  of  X  must  bo  increased  or  di- 
minished.    A  few  trials  will  give  the  correct  result. 

117.  Since  the  interval  of  time  I" — t  cannot  be  determined  with 
sufficient  accuracy  from  (65)  when  the  chord  x  ■:  very  small,  it 
becomes  necessary  to  cil'ect  a  further  transformation  of  this  eqtuition. 
Thus,  let  us  put 

q—q'=^QP,  a;  =  sin'  \s,  a/  =  sin'  \  d, 

and  we  shall  have 


P  ==  1  (a:  -  a;')  (  1  +  ?  (^  +  ^')  +  j^^'  C^-'  +  ^^•'  +  ^")  +  &c.  ). 


Now,  when  x  is  very  small,  we  may  put 

co8^e  =  cos]^, 


and  hence 


X  —  x'  =  sin''  j  e  —  sin' .{ '5 


sin'^e  — sin'^i 


4  cos'';}e 


which,  by  means  of  equations  (60),  becomes 


X  —  X  = 


8acos']c 
Therefore  we  have,  when  x  is  very  small, 


P  = 


If  we  put 


40a  cos'-|e 


^  (1  +  ',«  sin'  \s  -\-  8  0-  sin*  |e  +  &c.) 


,_t'  — P{r -{-/'— x)^ 

Q 


the  equation  (65)  becomes,  using  only  the  upper  sign, 

(r  +  r"  +  x)t  -  (r  +  r"-  x)t  =  6r„', 


(70) 
(72) 


which  is  of  the  same  form  as  (56)3.     Hence,  according  to  the  equa- 
tions (63)3  and  (66)3,  we  shall  have 


l/r  +  r" 


the  value  of  ft  being  found  from  Table  XI.  with  the  argument 


"?  = 


(73) 


(74) 


RELATION    HETAVEEN   TWO   PLACES   IX  THE   OUHrr. 


345 


It  remains,  thcrcforo,  simply  to  find  a  convenient  expn^sion  f^)r  r/, 
and  tlio  determination  of  x  is  I'tfeoted  by  a  pnx'ess  precisely  the  same 
as  in  the  special  case  of  parabolic  motion. 
Let  us  now  [)iit 

and  we  shall  have 

.-     cos'le/,    ,  2.8  .  ,,     ,  3.8.10  .  ,,    ,4.8.10.12  .  ,,    ,   „     \ 
^^Q-  \  1  +  -7"  «i"'^+    7  _  y    8inM£+-  7;y  ;j^     HmM=-+cto.  |, 


or,  substituting  for  Q  its  value  in  terms  of  sin  ^e, 

N=  1  +  33.  sin' le  +  ^Yo  sin*  ^  +  8%r6  si"«  ^'  +  &c>. 
Therefore^  if  we  put 


'^^-ia-Jj,^^^''-^'' 


(75) 


(76) 


the  expression  for  r^'  becomes 


ro'  =  '^-^V. 


(77) 


Tal)lc  XV.  gives  the  value  of  log  N  corresponding  to  values  of  8 
from  £  =  0  to  e  ==  60°. 

If  the  chord  H  is  given,  and  the  interval  of  time  t" —  t  is  required, 
we  compute  Ar„'  by  means  of  (76),  and,  having  found  rj  from 


,       xl/r  +  r" 
'»  =  ~~27x~' 

as  in  the  case  of  parabolic  motion,  wo  have 


\i  should  be  observed  that  although  equation  (76)  is  derived  for  the 
case  of  a  small  value  of  x,  yet  it  is  applicable  whcuever  t!ie  differ- 
ence e  —  5  is  very  small,  whatever  may  be  the  value  n'  x.  For 
orbits  which  differ  but  little  from  the  parabolic  form,  it  will  in  all 
cases  be  sufficient  to  use  this  expression  for  Ar^';  and  for  oases  in 
which  the  difference  between  e  and  d  is  such  that  the  assumption  of 
cos  ^£  =  cos  \d,  X  -{-  x'  =  2x,  &c.,  made  in  deriving  equation  (70),  does 


il 


340 


THEOIIKTICAL   ASTRONOMY. 


not  nfTord  llio  roquirod  iKicuracy,  we  may  compute  both  Q  niul  Q' 
direct ly,  iind  then  wo  have 


AT, 


;=i(i-^)o-+r"-H)i. 


(78) 


Tlic  vahics  of  the  Hu!tor  J  I  1  —  -  I  may  be  tabulated  direetly  with 

-J —  {IS  the  vertical  argument  and  j—  as  the  horizontal  argument; 

but  for  the  few  oases  in  which  the  value  of  iV  given  by  the  equation 
(7o)  is  not  sufliciently  accurate,  it  will  be  easy  to  compute  Q  and  Q' 
by  means  of  the  formuhe  (GO)  and  (08),  and  then  find  Ar^'  from  (78). 
Further,  wlu  n  there  is  any  doubt  as  to  the  accuracy  of  the  result 
given  by  (76),  for  the  final  trial  in  finding  x  from  r  +  r"  and  r^,  by 
means  of  the  equations  (73)  and  (74),  it  will  be  advisable  to  compute 
APu'  from  (78). 

It  apj)ears,  therefore,  that  for  nearly  all  the  cases  which  actually 
occur  the  determination  of  the  value  of  x,  corresiionding  to  given 

values  of  a  and  31=  — .  is  reduced  by  means  of  the  equation  (72)  to 

the  method  which  is  adoi)ted  in  the  case  of  parabolic  orbits. 

The  calculation  of  the  numerical  values  of  r  -j-  r"+  a  and  r  +  r"~K 
will  be  most  conveniently  cffoctcd  by  the  aid  of  addition  and  sul)- 
ti'action  logarithms.  If  the  tables  of  common  logarithms  are  used, 
we  may  first  compute 

sin  r  =  — I — r,, 
r  +  r 

and  then  we  liave 

r  +  r"  +  X  =  2  (r  +  r")  sin^  (45°  +  J^/), 
r  +  r"  —  X  =  2  (r  +  r")  cos'^  (45°  +  I/). 

118.  In  the  case  of  hyperbolic  motion,  the  semi-transverse  axis  is 
negative,  and  the  values  of  sin  Js  and  sin  ^d  given  by  the  equations 
(60)  bocome  imaginary,  so  that  it  Ls  no  longer  possible  to  compute 
the  inverval  of  time  from  r  +  r"  and  x  by  means  of  the  auxiliary 
angle  i  e  and  8.     Let  us,  therefore,  put 


sin'  ttS 


—  w' 


8in'^<5  = 


n'; 


then,  when  a  is  negative,  m  and  n  will  be  real.     Now  we  have 


and 


=  sin      V— 


7?l', 


^e  l/ — 1  =  logs  (cos  ^e  -f-  V^ — i  sin  ^s). 


RELATION   BKTWEEN    TWO   PLACES   IX   THE  ORBIT. 

Hence  wc  derive 


347 


«  =  2  sin     V  —  ,n*  =  -  y  —  log.  (l/ 1  +  m'  +  m), 


<J  =  2  sin  "'  ]/—  »'  =  -—-.  log,  (-[/i  -f  n'  +  Ji). 

Siil)stituting  those  values  in  the  ocjuation  (61),  nml  writing  —  a  m- 

t'toiul  of  (I,  .since 

sin  e  =  2m  V~l  •  l/l  -f  vi', 
wo  shall  have 


^  =  2m  l/l  +  w»  —  2  log,  (l/l  +  m'  +  m) 


Oi 


(2n  V'l  +  n'^  —  2  log.  (v/l  +  «'  +  ■«)), 


(79) 


the  upper  .sign  being  used  whoa  the  heliocentric  motion  is  less  than 
180°,  and  the  lower  .sign  when  it  is  greater  than  180°.  Therefore, 
if  we  compute  m  and  n  from 


m 


-4 


r  +  r"+x 


4a 


71 


4 


r  +  r" —  X 


4a 


(80) 


rcgfii'ding  the  hyperbolic  .'seini-transverse  axis  a  as  positive,  the  for- 
mula (79)  will  determine  the  interval  of  time  t'  =  k{t"  —  t). 

The  fir.st  two  terms  of  the  second  member  of  equation  (79)  may 
be  expressed  in  a  series  of  ascending  powers  of  «t,  and  the  last  two 
terms  in  a  series  of  ascending  powers  of  n.     Thus,  if  we  put 

log,  (l/l  -f-  Hi'''  +  m)  =  am  -f  (Jhi"  +  ym'  +  ^"'■*  +  &c., 
wo  get,  by  differentiation, 
1 


l/ 1  +  j/i'' 


.  =  a  +  2,3m  +  Srm'  +  4(5»i'  +  5s»i*  +  &c. ; 


and  since 


1-3 


1/1  + 


m 


=  =  l_i,n»  +  ^--— w* 


1  •  3  •  5 


we  have 

a  =  i,       /9  =  o,       r  =  -^-h 

Hence  we  obtain 


2-4-(J 


«5  =  0, 


rm'  +  &c., 


^  =  ^2^4'* 


3-5 


2  log.  (l/l  4-  ?tt'  +  «i)  =  2m  —  Jm'  +  i  •  |m»  —  4  |-^  m'  +  &c. 


¥ 


348 

We  have,  also, 


THEORETICAL  ASTRONOMY. 


1-3 


Therefore, 


2»i  VI  -\-  wi*  =  2»i  +  *'**  —  i'"'*  +  r-n  *"•'  —  &c 

4  •  b 


2m  V 1  +  «i^  —  2  loge  (V'  1  +  »i'^  +  m)  = 
and  similarly 


2n  Vl  +  n'  —  2  log„ (i/l  +  )i^  +  n)  = 


?i' 


^-^^•^--&c.).     ^«^) 


^2-l 


Substituting  these  values  iji  the  equation  (79),  and  denoting  the 
series  of  terms  enclosed  in  the  i)arentheses  by  Q  and  Q',  respectively, 
we  get 

6r'  =  Q  (r  +  r"  +  J*)'  =F  <'/  (r  +  r"  -  «)^  (83) 

which  is  identi(*al  with  equation  (65).  If  we  replace  w^  by  — sin-U 
and  n"  by  —  sin^iiJ  in  the  expressions  for  Q  and  Q',  as  given  by  (81) 
and  (82),  we  shall  have  the  exi)ressions  foi*  these  (quantities  in  terms 
of  sin  Is  and  sin  id,  respectively,  instead  of  sin  \e  and  sin  ]o  as  given 
by  the  equations  (62)  and  (63),  namely. 


(2  :- 1  +  f  •  i  siuMe  +  ^  1^1  slu*  ^.  +  ^  ^|;|  sin*:  ^H- &c., 

Q'  -  1  +  r  i  ^k^  +  I  \~^  sin* W  +  I  2t1vJ  "'^'  ^^  +  «^«- 


84) 


For  the  case  of  an  elliptic  orbit  it  is  most  convenient  to  use  the 
equations  (66)  and  (68)  in  finding  Q  and  Q' ;  but,  since  the  cases  of 
hyperbolic  motion  arc  rare,  while  for  those  which  do  occur  the  eocen- 
ti'icity  is  very  little  greater  thtm  that  of  the  parabola,  it  will  be  suf- 
ficient to  tubulate  (,>  directly  with  the  argument  m.  The  «anic  liiltlc, 
using  n  as  the  argument,  will  give  the  vniue  of  Q/.  Talkie  XA'I. 
gives  the  values  of  Q  corresponding  to  values  of  m  from  ri>  -  0  to 
m  =--  0.2. 

When  the  values  of  r  -f  r",  r',  and  a  are  given,  and  the  chord  '/i 
is  required,  we  may  compute  at,/  from  (78),  r,/  fi-om  (77),  and  finally 
H  from  (73). 

It  may  be  remarked,  also,  that  the  formulae  for  the  relation  between 
r',  )•-]-  ?•",  K,  and  a  sufliee  to  find  by  trial  the  value  of  a  wlien  r  +  ''" 
and  X  art!  given.     Hence,  in  the  computation  of  an  orbit  from  assumed 


EELATIOX    BETWEEN    TWO   PLACES   IN   THE   ORBIT. 


340 


values  of  J  and  J",  the  value  of  x  intiy  be  computed  iVoni  /•.  /•",  and 
u"  —  II,  and  then  a  may  be  found  in  the  manner  hero  indieated, 

If  wo  substitute  in  the  eqnations  (84)  the  values  of  sin  |£  and  sin  Id 
in  terms  of  r  -\-  r",  K',  and  a,  and  then  substitute  the  resulting  values 
of  Q  and  Q'  in  the  equation  (Go),  we  obtain 


1 


+  B§0  ^  (('•  +  r"  +  x)i  ::f  (r  +  r"  -  x)?)  +  &c,  ^^''^ 

(JL 

the  lower  sign  being  used  when  u"  —  u  exceeds  180°.  When  the 
eccentricity  is  very  nearly  equal  to  unity,  this  series  converges  with 
(Treat  i'a])idity.     In  the  case  of  hyperbolic  motion,  the  sign  of  a  must 

t)e  oliangod. 

119.  The  formuhe  thus  derived  for  the  determination  of  the  chord  x 
for  the  eases  of  elliptic  und  hyperbolic  orbits,  enable  us  to  correct  an 
uppioxiiuate  orbit  by  varying  the  semi-transverse  axis  a  and  the 
nuiu  J/ of  two  curtate  distances.  But  siui-e  the  formuhe  will  gene- 
riilly  be  applied  for  the  correction  of  approximate  parabolic  elements, 

or  those  which  ave  nearly  2)arabolic,  it  will  be  expedient  to  use  -  and 

3/ as  the  quantities  to  be  determined, 
lu  the  first  place,  we  compute  a  system  of  elements  from  M  and 

/=-;  and,  for  the  determination  of  the  auxiliary  quantities  pre- 

limiiKiry  to  the  calculation  of  the  values  of  r,  r",  and  X,  the  equa- 
tions (41).j,  (00)3,  and  (51)3  will  be  employed  when  the  ecliptic  is  the 
fiindiuuental  plane.  But  when  the  equator  is  taken  as  the  funda- 
menlal  ]>lane,  we  must  first  compute  y,  K,  and  G  by  means  of  the 
"'Illation-^  '>'*5):i-  Then,  by  a  process  entirely  aiialogous  to  that  by 
■>\vieh  the  equations  (47)3  and  (00)3  were  derived,  we  obtain 

h  cos  :  cos  (H—  a")  --=  M-  cos  (a"  —  a), 

h  00s  :  sin  ( ^  —  a'')  =---  sin  (a"  —  a),  (86) 

h  sin  C  =  M  tan  'J"  —  tan  d, 

frosn  which  to  find  II,  ^,  and  h  ;  and  also 

cos ^  =  cos  ? cos K cos (G  —  H)  -\-  sin  ? sin K,  (87) 

from  which  to  find  <f.  In  this  caf?e,  ^  and  //  will  be  referred  to  the 
etjuator  as  the  fundamental  plane.  The  angles  i/^  and  1^"  will  be 
obtained  from  the  equations  (102)3,  ^^  ^^'^i^i  equations  of  the  form 


350 


THEORETICAL   ASTRONOMY. 


of  (2G),  and  finally  the  auxiliary  quantities  A,  B,  B",  &c.  -svill  be 
obtained  from  (51)3,  writing  d  and  u"  in  place  of  /.9  and  ^",  respect- 
ively. 

As  soon  as  these  auxiliary  quantities  have  been  determined,  by 
means  of  (52)3  the  value  of  y.  must  be  found  which  will  exactly 
satisfy  equation  (65).     To  effect  this,  we  first  compute  e  from 


and,  if  it  be  required,  we  also  find  d  from 


sinA^  =  l/|/(r-{-r" 


«), 


using  aj)proximate  values  of  r  +  r"  and  x.  Then  we  find  Q  from 
(G6),  and  Ar^'  from  (7(j)  or  from  (78),  the  logarithms  of  the  auxiliary 
(][uantitics  i?^  and  N  being  found  by  means  of  Table  XV.  with  the 
argument  e.  The  value  of  r/  having  been  found  from  (77),  the 
equations  (73)  and  (74),  in  connection  with  Table  XI.,  enable  us  to 
obtain  a  closer  approximation  to  the  correct  value  of  x.  With  this 
we  compute  new  values  of  r  and  }•",  and  repeat  the  determination 
of  Jt.  A  few  trials  Avill  generally  give  the  correct  result,  and  these 
trials  may  be  facilitated  by  the  use  of  the  formula  (67)3.  It  will  he 
observed,  also,  that  Q  and  at,,'  are  very  slightly  changed  by  a  small 
v.!hange  in  the  values  of  r  +  r"  and  x,  so  that  a  repetition  of  the 
calculation  of  these  quantities  only  becomes  necessary  for  the  fnial 
trial  in  finding  the  value  of  x  which  completely  satisfies  the  equa- 
tions (52)3  and  (65).  When  the  value  of  a  is  such  that  the  values 
of  Q  and  iY  exceed  the  limits  of  Table  XV.,  the  equation  (61)  n)ay 
be  employed,  and,  in  the  case  of  hyperbolic  motion,  when  (^  and  Q' 
exceed  the  limits  of  Table  XVI.,  we  may  employ  the  com])lete  ex- 
prt.'ssion  for  the  time  r'  in  terms  of  m  and  n  as  given  by  (79). 
The  values  of  r,  r",  and  x  having  thus  been  found,  the  equations 


d  =  i/x»  — ^», 


d  -\-  g  cos  ip 

1:  ' 


p"=:-Mp, 


will  determine  the  curtate  distances  p  and  (>".     When  the  equator  is 
the  fundamental  plane,  we  have 


pj=  AcQ^  (?, 


J"  cos  .5". 


Froni  p,  p",  and  the  corresponding  geocentric  spherical  co-ordinates, 
the  radii-vectores  and  the  heliocentric  spherical  co-ordinates  /,  /",  6, 
and  b"  will  be  obtained,  and  thence  Q,,  i,  u,  u",  and  'the  leniaiuing 


VARIATION  OF  THE  SEMI-TRANSVERSE   AXIS. 


351 


clemonts  of  tlie  orbit,  as  already  illustrated.  In  the  case  of  elliptic 
motion,  if  we  compute  the  auxiliary  quantities  e  and  d  by  means  of 
the  equations  (60),  we  shall  have 

e  sin  \  (E"  -\-E)  =  ,.     !^'~''     ... 

e  cos  i  (E"  +  E)  =  cos  i  (e  +  <5), 

from  which  c  and  \{E" -\-  E)  may  bo  found,  and  hence,  since 
\{E"  —  E)  ^\{t  —  d),  we  derive  E  and  E".  The  values  of  q  and 
V  may  then  be  found  directly  from  these  and  quantities  already 
obtained.     Thus,  the  last  of  equations  (43)i  gives 

cos  \v cos  \E  cos  {v" cos  \E" 

V~q    "     V'r   '  Vq  l//' 

^lultiplying  the  first  of  these  expressions  by  sin  Jy",  and  the  second 
by  —  sin  |t',  adding  the  products,  and  reducing,  we  obtain 

sin  2  (v"  —  v)  sin  -] y cos  I  {v"  —  v)  cos  ^E      cos  .\ J5"' 


Vq 

Tlierefore,  we  shall  have 


Vr 


--=sin-.Vy : 
Vq        ' 

1  , 

-7-  cos  SV  ■■ 

Vq        ' 


COS  IE 


cos  ^E 


ITT" 


l/r"sin^(tt"— «) 


(88) 


Vr  tan  A  (tt"—  ti) 
COS  IE 

from  which  q  and  v  may  be  found  as  soon  as  cos  \E  and  cos  \E"  are 
known.  In  the  case  of  parabolic  motion  the  eccentric  anomaly  is 
equal  to  zero,  and  these  equations  become  identical  with  (92)3.  The 
angular  distance  of  the  perihelion  from  the  ascending  node  will  be 
obtained  from 

Since  r^=a  —  ae  cos  J?,  and  (j*  =  a(l  —  c),  we  have 


C09>E-- 


1  — 

1  — 


and  hence 


1  — 


t-\ 


cos»iE=l 


cos'A£"=l 


9 

» 

"1-1 

Jl 


(89) 


352 


THEORETICAL   ASTRONOMY. 


When  the  eccentricity  is  nearly  equal  to  unity,  the  value  of  q  given 
by  apj>roxiniate  elements  will  be  sufficient  to  compute  cos^E  and 
cos^E"  by  moans  of  these  e(piations,  and  the  results  thus  derived 
will  be  substituted  in  the  equations  (88),  from  which  a  new  value  of 
(J  results.  If  this  should  differ  considerably  from  that  used  in  com- 
puting cos^E  and  cosJJS'",  a  repetition  of  the  calculation  will  give 
the  correct  result. 

In  tiie  case  of  hyperbolic  motion,  although  E  and  E"  are  imagi- 
nary, we  may  compute  the  numerical  values  of  cos^E  and  coa^E" 
from  the  c([uations  (89),  regarding  a  as  negative,  and  the  results  will 
be  used  for  the  corresponding  quantities  in  (88)  in  the  computation 
of  7  and  v  for  tlie  hyperbolic  orbit. 

Next,  we  compute  a  second  system  of  elements  from  3/and/+  8f, 
and  a  third  system  from  3f-\-d3Ian(lf,8fsim\  o3/ denoting  the 
arbitrary  increments  assigned  to  /  and  M  respectively.  The  com- 
parison of  these  three  systems  of  elements  with  addit'-  nal  observed 
places  of  the  comet,  will  enable  us  to  form  the  equations  of  condition 
for  the  determination  of  the  most  probable  values  of  the  corrections 
AiJ/and  a/ to  be  ai)plied  to  J/ and /respectively.  The  formation  of 
these  equations  is  effected  in  precisely  the  same  manner  as  in  the  case 
of  the  variation  of  the  geoceutiiC  distances  or  of  ^  and  /,  and  it  does 
not  require  any  further  illustration.  The  final  elements  will  be  ob- 
tained from  J/+  a3/,  and/+  a/,  either  directly  or  by  interpolation. 
We  may  remark,  further,  that  it  will  be  convenient  to  use  log  Ju  as 
the  quantity  to  be  corrected,  and  to  express  the  variations  of  log  M 
in  units  of  the  last  decimal  place  of  the  logarithms. 

When  the  orbit  differs  very  little  from  the  parabolic  form,  it  will 
be  most  exjieditious  to  make  two  hypotheses  in  regard  to  31,  putting 

in  each  case  -  =  0,  and  only  compute  elliptic  or  hyperbolic  elements 

in  the  third  hypothesis,  for  which  we  use  il/and  f=3f.  The  first 
and  second  systems  of  elements  will  thus  be  parabolic. 

120.  Instead  of  il/and  -  wc  may  use  J  and  -  as  the  quantities  to 

be  corrected.  In  this  case  we  assume  an  approximate  value  of  J  by 
means  of  elements  already  known,  and  by  means  of  (96)3,  (98)3.  (102)3, 
and  (103)3,  we  compute  the  auxiliary  quantities  C,  B,  B",  etc.,  il- 
quired  in  the  solution  of  the  equations  (104)3.  We  assume,  also,  an 
approximate  value  of  J"  and  com[)ute  the  corresponding  value  of  /•", 
the  value  of  r  having  been  already  found  from  the  assumed  value  of 
J.     Then,  by  trial,  we  find  the  value  of  x  which,  in  connection  with 


EQUATIONS  OF  CONDITION'. 


353 


the  assumed  value  of  -.  will  satisfy  the  equations  (104)3  ^^^*^  (^^)  or 
(61).     The  corresponding  value  of  J"  is  given  by 


A"^e±Vx'—C\ 

When  J"  has  thus  been  determined,  tb.e  heliocentric  places  will  be 
obtained  by  means  of  the  equations  (106)3  ^^^^  (10~)3,  and,  finally, 
the  corresponding  elements  of  the  orbit  will  be  computed.  If  the 
ecliptic  is  taken  as  the  fiuidamental  plane,  wo  put  D  =  0,  vl  =  O, 
and  write  ?.  and  /9  in  place  of  a  and  d  respectively. 

If  we  now  compute  a  second  system  of  elements  from  J  +  o  J  and 

/=  -,  and  a  third  system  from  J  and/+  df,  the  comparihon  of  the 

three  systems  of  elements  with  additional  observed  places  will  furnish 
the  equations  of  condition  for  the  determination  of  the  corrections 

A  J  and  a/  to  be  applied  to  J  and  -  respectively. 

When  the  eccentricity  is  very  nearly  equal  to  unity,  we  may  as- 
sume/=0  for  the  first  and  second  hypotheses,  and  only  compute 
elliptic  or  hyperbolic  elements  for  the  third  hypothesis. 

121.  The  comparison  of  the  several  observed  places  of  a  heavenly 
hotly  with  opo  of  the  three  systems  of  elements  obtained  by  varying 
the  two  quantities  selected  for  correction,  or,  when  the  required  dif- 
ferential coefficients  are  known,  with  any  other  system  of  elements 
sucli  that  the  squares  and  products  of  the  corrections  may  be  neg- 
lected, gives  a  series  of  equations  of  the  form 

mx  -f-  ny  ^=p, 
m'x  -{-  n'y  =  r>',  &c., 

in  M-hich  x  and  y  denote  the  final  corrections  to  be  applied  to  the  two 
assumed  f[uantities  respectively.  The  combination  of  these  equations 
which  gives  the  most  probable  values  of  the  unknown  quantities,  is 
effected  according  to  the  method  of  least  squares.  Thus,  we  multiply 
Oiicli  equation  by  the  coefficient  of  x  in  that  ecjuation,  and  the  sum 
of  all  the  equations  thus  formed  gives  the  first  normal  equation. 
Then  we  multiply  each  equation  of  condition  by  the  coefficient  of  y 
in  that  equation,  and  the  sum  of  all  the  j)roducts  gives  the  second 
normal  equation.     Let  these  equations  be  expressed  thus : — 

[mwi]  X  -f-  [inii]  y  =  Imp"], 

[inn']  X  +  [/tji]  y  ^  [np], 

23 


\\ 


354 


THEOKETICAL   ASTRONOMY. 


in  which  [»i»i]-— m^+m"'+«i"^+<S:c.,  [inn'\^=^mn-\-m'n'-\'in"n"-r&.Q., 
and  similarly  for  the  other  terms.  These  two  final  eqnatiojis  givo, 
by  elimination,  the  most  probable  valnes  of  x  and  y,  namely,  those 
for  which  the  sum  of  the  squares  of  the  residuals  will  be  a  mininuun. 
It  is,  however,  often  convenient  to  determine  x  in  terms  of  y,  or  y 
in  terms  of  x,  so  that  we  may  lind  the  influence  of  a  variation  of  one 
of  the  unknown  quantities  on  the  differetices  between  computation 
and  observation  when  the  most  probable  value  of  the  other  unknown 
quantity  is  used.  Thus,  if  it  be  desired  to  find  x  in  terms  of  y,  the 
most  probable  value  of  x  will  be 


[mn'\ 

— W. 

[inni] '' 


If  we  substitute  this  value  of  x  in  the  original  equations  of  condition, 
the  remaining  differences  between  computation  and  observation  will 
be  expressed  in  terms  of  the  unknown  quantity  y,  or  in  the  form 


A<?  =  7Ho  +  n^y. 


(90) 


Then,  by  assigning  different  values  to  y,  we.  may  find  the  correspond- 
ing residuals,  and  thus  determine  to  wdiat  extent  the  correction  y  may 
be  varied  without  causing  these  residuals  to  surpass  the  limits  of  the 
probable  errors  of  observation. 

In  the  determination  of  the  orbit  of  a  comet  there  must  be  more 
or  less  uncertainty  in  the  value  of  «,  and  if  y  denotes  the  correction 

to  be  applied  to  the  assumed  value  of  -,  we  may  thus  determine  the 

probable  limits  within  which  the  true  value  of  the  periodic  time 
must  be  found.  In  the  case  of  a  comet  which  is  identified,  by  the 
similarity  of  elements,  with  one  which  has  previously  appeared,  if 
we  compute  the  system  of  elements  which  Avill  best  satisfy  the  series 
of  observations,  the  supposition  being  made  that  the  comet  has  per- 
formed but  one  revolution  around  the  sun  during  the  intervening 
interval,  it  will  be  easy  to  determine  whether  the  observations  are 
better  satisfied  by  assuming  that  two  or  more  revolutions  have  been 
completed  during  this  interval.  Thus,  let  T  denote  the  periodic 
time  assumed,  and  the  relation  between  T  and  a  is  expressed  by 


T= 


2-al 


in  which  t:  denotes  the  semi-circumference  of  a  circle  whose  radius 


ORBIT  OF  A   COMET. 


866 


is  unity.     Let  the  periodic  time  con'csponding  to  -  +  w  bo  denoted 

m  U 

by  —  ;  then  we  shall  have 


*       a 


1 

a 


and  the  equations  for  the  residuals  arc  transformed  into  the  form 

A<?  =:(»»„-«  J)  +«oM  (91) 

If  we  now  assign  to  z,  successively,  the  values  1,  2,  3,  &c,,  the  re- 
siduals thus  obtiiined  will  indicate  tiie  value  of  z  which  best  satisfies 
the  series  of  observations,  and  hence  how  many  revolutions  of  the 
comet  have  taken  2)lace  during  the  interval  denoted  by  T, 

122.  In  the  determination  of  the  orbit  of  a  comet  from  three  ob- 
served places,  a  hypothesis  in  regard  to  the  semi-transverse  axis  may 
with  facility  be  introduced  simultaneously  with  the  computation  of 
the  parabolic  elements.  The  numerical  calculation  as  far  as  the  form- 
ation of  the  equations  (52)3  ^^'^^  ^^  precisely  the  same  for  both  the 
parabolic  and  the  elliptic  or  hyperbolic  elements.  Then  in  the  one 
case  we  find  the  values  of  r,  /■",  and  z  which  will  satisfy  equation 
(56),,  and  in  the  other  case  we  find  those  which  will  satisfy  the  equa- 
tion (65),  as  already  explained.     From  the  results  thus  obtained,  the 

two  systems  of  elements  will  be  computed.     Let /=->  then  in  the 

ease  of  the  system  of  parabolic  elements  we  havc/=0,  and  the  com- 
parison of  the  middle  place  with  these  and  also  with  the  ellnitic  or 
hyperbolic  elements  will  give  the  value  of 


do 


O.-Or 


ill  which  d^  denotes  the  geocentric  spherical  co-ordinate  computed 
from  the  parabolic  elements,  and  6.^  that  computed  from  the  other 
system  of  elements.  Further,  let  tid  denote  the  difference  l)etween 
oomputation  and  observation  for  the  middle  place,  and  the  correction 
to  he  applied  to  /,  in  order  that  the  computed  and  the  observed 
values  of  d  may  agree,  will  be  given  hy 


do 
df 


£,f  j-A0  =  O. 


Honcc,  the  two  observed  spherical  co-ordinates  for  the  middle  place 
will  give  two  equations  of  condition  from  which  a/  may  be  found, 


356 


THEORETICAL   ASTRONOMY. 


and  tlio  corresponding  elements  will  bo  those  which  best  represent 
the  observations,  assuming  the  adopted  value  of  3i  to  be  correct. 

123.  The  first  dotcrmination  of  the  approximate  elements  of  the 
orbit  of  a  comet  is  most  readily  eflected  by  adopting  the  ecliptic  as 
the  fundamental  plane.  In  the  subsequent  correction  of  these  ele- 
ments, by  varying  -  and  31  or  J,  it  will  often  be  convenient  to  use 

the  cfiuator  as  the  fundamental  plane,  and  the  first  assumption  in 
regard  to  JI  will  be  made  by  means  of  the  values  of  the  distances 
given  by  the  approximate  elements  already  known.  But  if  it  bo 
desired  to  compute  31  directly  from  three  observed  places  in  reference 
to  the  ccjuator,  without  converting  tiie  right  ascensions  and  declina- 
tions into  longitudes  and  latitudes,  the  re(piisite  formulae  may  be 
derived  by  a  process  entirely  analogous  to  that  employed  when  the 
curtate  distances  refer  to  the  ecliptic.  The  case  may  occur  in  which 
only  the  right  ascension  for  the  middle  place  is  given,  so  that  the 
corresponding  longitude  cannot  be  found.  It  will  then  be  necessary 
to  adopt  the  equator  as  the  fundamental  plane  in  determining  a 
system  of  parabolic  elements  by  means  of  two  complete  observations 
and  this  incomplete  middle  place.  If  we  substitute  the  expressions 
for  the  heliocentric  co-ordinates  in  reference  to  the  equator  ui  the 
equations  (4)3  and  (0)3,  we  shall  have 

0  =  n  (p  cos  o  —  R  cos  D  cos  A)  —  (/  cos  o'  -  R'  cos  U  cos  A') 

+  7i"  y  sin  a"—  R"  cos  D"  cos  A"), 

0  =  « (/)  sin  a  —  i?  cos  D  sin  A)  —  (p'  sin  a'—  R'  cos  D'  sin  A')  (92) 

+  n"  (p"  sin  a"—  R"  .03  D"  sin  A"), 

0  =  )i  (p  tan d  —  RsinD)  —  (p'  tan  <5'—  R'  sin  D') 

+  n"ip"tanr  —  R"smD"l 

in  which  f>,  />',  p''  denote  the  curtate  distances  with  respect  to  the 
equator,  A,  A',  A"  the  right  ascensions  of  the  sun,  and  D,  D',  B" 
its  declinations.  These  equations  correspond  to  (6)3,  and  may  be 
treated  in  a  similar  manner. 

From  the  first  and  second  of  equations  (92)  we  get 

0=^11  {p  sin  (a'— a)  —  i?  cos  Z)  sin  {o!—A))  +  R'  cos  D'  sin  (o'— ^') 

—  n"  (/>"  sin  (a"—  a')  +  R"  cos  D"  sin  (a'-  ^1")), 
and  hence 


3/-= 


sin  (a' —  a) 


n 


(93) 


p        IV    sm(,o    — a') 
nR  cos  D  sin  (a'— ^)— i2'  cos  D'  sin  {o:—A')+n"R"  cos  D"  sin  (a'— ^") 


pn"  sin  (tt"  —  a') 


VARIATION  OF  TWO   RADII-VECTORE8. 


357 


This  formula,  being  independent  of  the  declination  <V,  may  be  used 
to  compute  31  when  only  the  right  ascension  for  the  middle  place  is 
given.  For  the  first  assumption  in  the  case  of  an  unknown  orbit, 
we  take 

f—f     sin  (o'j-  a) 

if—t'  sin  (a"    ~ 


M. 


a) 


and,  by  means  of  the  results  obtained  from  this  hypothesis,  the  com- 
plete expression  (93)  may  be  computed.  By  a  process  identical  with 
that  employed  in  deriving  the  ef^uation  (36)3,  we  derive,  from  (93), 
the  expression 


„ n      sin  (a'  —  a) 

''    ~^^'sfn(a'"'-~a'y 


and,  putting 

n      sin  f  tt'  —  a) 


(94) 
1  \i?'cos/)'9in(a'  — ^') 


sin  (tt"  —  tt') 


M,-- 


n"'  sin  (a"— a')' 


n"    zt' 


^==^-^-^•7'-^'  +"^~siirra^^tty~-7 


\r"      EV' 


we  have 


M='-=ALF. 


(95) 


The  calculation  of  the  auxiliary  quantities  in  the  equations  (52)3 
will  be  effected  by  means  of  the  formuhe  (96)3,  (86),  (87),  (102),,  and 
(51)3.  The  heliocentric  places  for  the  times  t  and  t"  will  be  given 
by  (106)3  '^"^^  (107)3,  ^"^^  ^''0'^  these  the  elements  of  the  orbit  will 
be  Ibiuid  according  to  the  process  already  illustrated. 

124.  The  methods  already  given  for  the  correction  of  the  approxi- 
mate elements  of  the  orbit  of  a  heavenly  body  by  means  of  additional 
ot).sorvations  or  normal  places,  are  those  which  will  generally  be 
a])plied.  There  are,  however,  modifications  of  these  which  may  be 
iidviintageous  in  I'are  and  special  cases,  and  w^hich  will  readily  suggest 
themselves.  Thus,  if  it  be  desired  to  correct  approximate  elements 
by  varying  two  radii-vectores  /•  and  r",  we  may  assume  an  api)roxi- 
mate  value  of  each  of  these,  and  the  three  equations  (88)1  will  con- 
tain only  the  three  unknown  quantities  J,  h,  and  /.  By  elimination, 
these  unknown  quantities  may  be  found,  and  in  like  manner  the 


358 


TIIEOKETICAI.   ASTRONOMY. 


values  of  J",  h",  and  /".     It  will  be  most  convenient  to  compute 
the  angles  ^  and  i^",  and  then  find  z  and  z"  from 


sin? 


R  sin  4- 


.     „      R"  sin  4" 
sni3   = ,T — I 


or,  putting  a?=^i^  —  K'  sin^i//,  and  x""^  =  ?•"*  —  iR"^  sin";//",  from 


R  sui  4 
tans  = , 

X 


tan  3  = ,, — . 


The  curtate  distances  will  be  given  by  the  equations  (3),  and  the 
heliocentric  spherical  co-ordinates  by  means  of  (4),  writing  r  in  place 
of  a.  From  these  ti"  —  u  may  be  found,  and  by  means  of  the  values 
of  r,  r",  and  u"  —  u  the  determination  of  the  elements  of  the  orbit 
may  bo  completed.  Then,  assigning  to  r  an  increment  dr,  we  com- 
pute a  second  system  of  elements,  and  from  r  and  /"  -f  S>'"  a  third 
system.  The  comparison  of  these  throe  systems  o  lements  with  an 
additional  or  intermediate  observed  place  will  furnish  the  equations 
for  the  determination  of  the  corrections  Ar  and  ^r"  to  be  ap[)lie(l  to 
'/•  and  r",  res})ectively.  The  comparison  of  the  middle  place  may  be 
made  with  the  observed  geocentric  spherical  co-ordinates  directly,  or 
with  the  radius-vector  and  argument  of  the  latitude  computed  directly 
from  the  observed  co-ordinates;  and  in  the  same  manner  any  number 
of  additional  observed  i)lace8  may  be  employed  in  forming  the  e(pi:i- 
tions  of  condition  for  the  determination  of  Ar  and  Ay". 

Instead  of  r  and  r",  we  may  take  the  projections  of  these  radii- 
vectores  on  the  plane  of  the  ecliptic  as  the  quantities  to  be  corrected. 
Let  these  projected  distances  of  the  body  from  the  sun  be  denoted 
by  }•„  and  v,/',  respectively ;  then,  by  means  of  the  equations  (88)i, 

we  obtain 

R  sin  U  —  O) 


sin(^  —  ^) 


(96) 


from  which  /  may  be  found ;  and  in  a  similar  manner  we  may  find 
l".     If  we  put 


we  have 


tan  (Z  —  A) 


R'shi'i^  —  Q), 
i?sin(A  — O) 


(97) 


liet  S  denote  the  angle  at  the  sun  between  the  earth  and  the  place 
of  the  planet  or  comet  projected  on  the  plane  of  the  ecliptic ;  then 
we  shall  have 


v'AniATION    OF   TWO    UAnil-VKCTonKH. 


359 


6": 

P  ■■ 


:  180°  +  O  -  /, 

sin  (7  — A)    ' 


ami 


tan  b 


p  tan  /S 


(98) 


(99) 


by  means  of  wliioh  the  lieliooentrie  latitudes  b  and  b"  may  be  found. 
The  calculation  of  the  elements  and  the  correction  of  /•„  and  /•„"  are 
then  ellected  as  in  the  case  of  the  variation  of  r  and  /•". 

In  the  case  of  parabolic  motion,  the  eccentricity  bein^  known,  wc 
may  take  r^  and  T  as  the  quantities  to  be  corrected.  If  we  assume 
approximate  values  of  these  elements,  r,  r',  r",  and  r,  r',  v"  will  be 
given  i  nulled  lately.  Then  from  r,  /•',  /•"  and  the  observed  spherical 
co-urdinatos  of  the  body  Me  may  compute  the  valuts  of  u" — u'  and 
1/'  —  i(.  In  the  same  manner,  by  means  of  the  observed  places,  we 
compute  the  angles  »/"— u'  and  u' — n  correspond ing  to  ii  +  oij  and  7', 
and  to  r/ and  T -\  ol\  d<j  and  ^57' denoting  the  arbitrary  increments 
assigned  to  q  and  1\  respectively.  The  comparison  of  the  helio- 
centric motion,  during  the  intervals  t"  —  t'  and  t'  —  f,  thus  obtained, 
in  the  case  of  each  of  the  three  systems  of  elements,  from  the  ob- 
served  geocentric  })laces  with  the  corresponding  results  given  by 


u 


u'  =  v" 


V, 


u  —  u 


enables  us  to  form  the  equations  by  which  we  may  find  the  cor- 
rections Afy  and  A 2^  to  be  applied  to  the  assumed  values  of  q  and  T, 
respectively,  in  order  that  the  values  of  «"  —  w' and  u'  —  u  computed 
by  means  of  the  observed  })laces  shall  agree  with  those  given  by  the 
true  anomalies  computed  directly  from  q  and  T. 


360 


TIIEOUKTICAL   AHTUONOMY. 


CHAPTER  Vri. 


MirritOD  OF  I.KA8T  SfirAllKS,  TIIKOUY  OK  TIIK  fOMIUSATION  OF  OBSEHVATIONS,  AND 
i.l.  rviiwiy  iTlox  (,K  IHK  MOST  I'ltOIJAIlJ-K  SYSTKM  OF  KLEMEXTS  I'llOM  A  !iEUII>! 
OF   OIISEIIVATIONS. 

1 25.  Wli  KX  the  olcnicntH  of  the  orbit  of  a  heavenly  body  an;  known 
to  such  a  decree  of  approximation  that  the  scpiares  and  jjroduets  of 
the  corrections  which  should  be  applied  to  them  may  be  nejjjlected, 
by  computing  the  partial  differential  coefficients  of  these  elements 
with  respect  to  each  of  the  obs(;rved  spherical  eo-ordinates,  \\v  may 
form,  by  means  of  the  differences  between  computation  antl  observa- 
tion, the  ccjuations  for  the  determination  of  these  corrections.  Three 
complete  observations  will  furnish  the  six  efpiations  required  for  the 
determination  of  the  corrections  to  be  applied  to  the  six  elements  of 
the  orbit ;  but,  if  more  than  three  complete  places  are  piven,  the 
number  of  ecjuatious  will  exceed  the  number  of  unknown  (piantities, 
and  the  problem  will  be  more  than  determinate.  If  the  observed 
places  were  absolutely  exact,  the  cond)inatiou  of  the  equations  of 
condition  in  any  manner  whatever  would  furnish  the  values  of  these 
corrections,  such  that  each  of  these  ecpiations  would  be  completely 
satisfied.  The  conditions,  however,  which  present  themselves  in  the 
actual  correction  of  the  elements  of  the  orbit  of  a  heavenly  body  by 
means  of  given  observed  j)laees,  are  entirely  different,  ^^'hen  the 
observations  have  been  corrected  for  all  known  instrumental  errors, 
and  when  all  other  known  corrections  have  been  duly  applied,  there 
still  remain  those  accidental  errors  which  arise  from  various  causes, 
such  as  the  abnormal  condition  of  the  atmosphere,  the  imperfections 
of  vision,  and  the  imperfections  in  the  performance  of  the  instrument 
employed.  These  accidental  and  irregular  errors  of  observation  camiot 
be  eliminated  from  the  observed  data,  and  the  equations  of  condition 
for  the  determination  of  the  corrections  to  be  applied  to  the  elements 
of  an  approximate  orbit  cannot  be  completely  satisfied  by  any  system 
of  values  assigned  to  the  unknown  quantities  unless  the  number  of 
equations  is  the  same  as  the  number  of  these  unknown  quantities. 
It  becomes  an  important  problem,  therefore,  to  determine  the  par- 
ticular combination  of  these  equations  of  condition,  by  means  of  which 


METHOD   OF    I.KAHT   SQITARKS. 


361 


tlu'  resulting  values  of  the  tuiknown  quantities  will  he  those  whieli, 
\vhil(^  they  do  not  eoinplctciy  siitisly  the  several  equations,  will  allord 
the  hijjjhest  dejrree  of  prohaliility  in  favor  of  their  neeuraey.  It  will 
be  of  interest  also  to  determine,  as  far  as  it  may  he  possihlc,  the 
degree  of  aeeuraey  whieh  may  hu  attributed  to  the  separate  results. 
But,  in  order  to  simi)lify  the  more  general  problem,  in  whitih  the 
quantities  sought  are  determined  indireetly  by  observation,  it  will  bo 
expedient  to  eonsider  first  the  sim[)Ier  case,  in  whieh  a  single  (piantity 
is  obtained  directly  by  observation. 

120.  If  the  accidental  errors  of  observation  could  be  obviated,  the 
(liflerent  determinations  of  a  magnitude  directly  by  observation  would 
be  iilentieal ;  but  since  this  is  impossible  when  an  extreme  limit  of 
precision  is  sought,  we  adopt  a  victtn  or  average  value  to  be  derived 
from  the  separate  results  obtained.  The  adopted  value  may  or  nuiy 
not  agree  with  any  individual  result,  since  it  is  only  necessary  that 
the  residuals  obtained  by  comparing  the  adopted  value  with  the 
observed  values  shall  be  such  as  to  make  this  adopted  value  the  7and 
jtrubalilr  value.  It  is  evident,  from  the  very  nature  of  the  case,  that 
wc  approach  here  the  confines  of  the  unknown,  and,  before  wc  pro- 
ceed further,  something  additi(tnal  must  be  assumed. 

However  irregular  and  uncertain  the  law  of  the  accidental  errors 
of  observation  may  be,  we  may  at  least  assume  that  small  errors  arc 
more  probable  than  large  errors,  and  that  errors  surpassing  a  certain 
limit  will  not  occur.  Wc  may  also  assume  that  in  the  case  of  a  large 
number  of  observations,  errors  in  excess  will  occur  as  frequently  as 
errors  iu  defect,  so  that,  in  general,  positive  and  negative  residuals 
of  equal  absolute  value  are  equally  ]>robab]e.  It  ai)pcars,  therefore, 
that  the  relative  frequency  of  the  occurrence  of  an  accidental  error  J 
in  the  observed  value  will  depend  on  the  magnitude  of  this  error, 
and  may  be  expressed  by  <f  (J).  This  function  will  also  express  the 
prohal)ility  of  an  error  J  in  an  observed  value.  At  the  limit  beyond 
which  an  error  of  the  magnitude  J  can  never  occur,  we  must  have 
^(J)  =  0:  when  J  =  0,  the  value  of  (p  (J)  must  l)e  a  maximum,  and 
for  equal  positive  and  negative  values  of  J  the  values  of  (p{-i)  must 
be  the  same.  Hence,  in  a  given  series  of  observations,  the  number  m 
of  observations  being  supposed  to  be  large,  the  number  of  times  in 
Mliich  the  error  J  occurs  will  be  expressed  by  mip{d),  and  the  number 
of  times  in  which  the  error  A'  occurs  will  be  expressed  by  mtp  (J'),  so 
that  we  shall  have 

m  =  mip  {S)  -f  mf  (J')  -f  vif  (-4")  +  &c., 


'•mi^fmmmmmmm^immmmmammmmmmmmmmm 


m2 


or 


TIIEORKTICAL    ASTRONOMY. 


2V  (J)  =---]. 


Tlie  sum  2'  must  ho  taken  between  the  limits  for  wliieli  the  ao('id(Mital 
errors  of  observation  arc  considered  possible}  Init  since  the  assignment 
of  these  limits  is,  in  a  certain  sense,  arbitrary,  we  ninst  evidently 
have 


s^ 


A  —  —  CO 


(1) 


y.. 


the  value  of  <p  (J)  being  absolutely  zei'o  for  the  limits  +  oo  and 

Within  any  given  limits  there  are  an  infinite  number  of  values, 
any  one  of  whieli  may  j)ossibly  be  the  true  value  of  J,  and  hence 
the  number  of  the  functions  e.\|:)ressed  by  <f  (J)  must  be  infmite. 
The  ]n-ohal)ility  of  an  error  J  is  expressed  by  <f  {J),  and  will  be  the 
same  as  the  i)robability  tliat  the  error  is  contained  within  the  limits  J 
and  J  :  dJ.  The  latter  is  expressed  by  the  sum  of  all  the  lunetious 
f  (J)  between  the  limits  J  and  J  +  dJ,  or  by 

V{J)dJ. 

We  conclude,  therefore,  that  the  probability  thi't  an  error  falls  between 
the  liuiits  a  and  h  is  expressetl  by  the  integral 

f<p{J)dJ, 

a 

and  this  integral,  taken  so  as  to  include  all  possible  accidental  errors 
of  observation,  is,  according  to  equation  (1), 


(2) 


According  to  the  theory  of  probabilities,  the  probability  that  ilic 
errors  J,  J',  &c.  occur  simultaneously  is  equal  to  the  continued  pro- 
duct of  thc!  probabilities  of  the  occurrence  of  these  errors  separately. 
Let  /*  denote  the  probal)ility  that  tliesc  errors  occur  at  the  same  time 
in  tiie  given  scries  of  observed  values,  and  svc  liavc 


P=:.^(J).^(J').^(J")... 


(3) 


The  most  probalile  value  of  the  quantity  sought,  which  we  will  dc- 
n,)te  by  x,  must  evidently  be  that  which  makes  Pa  maximum.     If 


METHOD  OF   LEAST   SQITAUES. 


303 


we  tako  tlie  logarithms  of  both  nionibors  of  (jciiiation  (3),  and  dUfor- 
entiato,  the  condition  of  a  niaxiinuni  gives 


_  rflogy(J)      rf  J  ^  lorr  C  {  J' ) 

""      dJ        (7.C  "^ "   V/J' 


f/J' 


+  ttc. 


(4) 


Lot  »,  i(',  /(",  ifce,  l)e  the  observed  vahies  of  x,  and  m  the  number  of 
ol)!*ervations  ;  then  \vc  liave 

J  =  71  —  X,  J'  =  n'  —  X, 

and  hence 


J"  =  n"  -  X,  &c., 


dJ 
(ix 


dx 


dr 

dx 


Tiierefore  the  equation  (4)  becomes 
0 


d  log  <p  in  —  x)       d  log  <p  Cn'  —  ^)  _,    p^ 

d{n  —  x)       "^ rf(/i'  — a-)'      ■^*^''- 


(/3) 


This  etfuation  will  serve  to  determine  the  value  of  x  as  soon  as  the 
liMii\  of  the  function  svnibolized  bv  c-  is  known.  It  becomes  neces- 
siuy,  therefore,  to  make  some  further  as.-^umption  in  regard  to  the 
errors  J,  J',  J",  etc,,  in  order  that  the  form  of  this  function  may  be 
determined;  and,  although  the  hypothesis  which  presents  itself  gives 
directly  the  most  probable  value  of  .r,  since  the  function  <p{J)  is  sup- 
posed to  be  general,  we  nuxy  thus,  by  the  special  case,  determine  the 
form  of  this  function;  and  the  result  will  be  applicable  when,  instead 
of  tlio  value  of  a  single  quantity,  it  is  required  to  find  the  most  pro- 
t)!i!)le  values  of  several  unknown  <piantities  determined  indirectly  by 
oliservation. 

127.  The  principle  may  bo  received  as  an  axiom,  that  when  a 
series  of  ol)served  values  of  a  quantity  is  given,  if  the  circumstances 
under  which  the  separate  observations  were  made  are  similar,  so  that 
there  is  no  reason  for  preferring  one  result  to  another,  the  most  j)ro- 
huhle  value  of  the  (piantity  sought  is  the  arithnutical  mean  of  the 
several  results.     Hence  we  have 


«:= 


Ji  +  n'  +  »i"+.... 


m 


m  being  the  number  of  observed  values.     This  expression  gives 

0  =  («  —  x)  +  (n'  —  x)  +  in"  —  x)-\-  &c.,  (6) 

from  which  it  appears  that  the  algebraic  sum  of  the  residuals  is  equal 
to  v.ero.     The  equatiou  (5)  may  be  written 


364 


THEORETICAL  ASTRONOMY. 


^  \n  —  x)  d in  —  X)  (/t  —  x)  d  {ii  —  x)    ' 


and  the  comparison  of  this  with  (6)  shows  that 
d  log  <p(n  —  x)    __     d  log  tp  (n!  —  x) 


k, 


(7) 


(n  —  x)  d  in  —  z)       (n'  —  x)  d  {n'  —  x ) 

k  being  a  constant  quantity.     Hence  we  derive 

d  log,  f!  (J)  :=  kJ  dJ, 

the  integration  of  which  gives 

\og,viJ)^\kJ'  +  \og,c, 

logcC  being  the  constant  of  integration.     From  this  equation  tliero 
results 


<p  (J)  =  ce 


MIA4 


(8) 


in  which  c  is  the  base  of  Naperian  logarithms.  Since  ^(J)  diminishes 
as  J  increases,  the  quantity  k  must  be  essentially  negative,  and  if  we 
put  Ik  =^  —  /r,  we  shall  have 


-A'A^ 


If  we  substitute  this  value  of  <f{J)  in  the  equation  (2),  we  have 

-f-oo 

e         tZJ  =  1, 
or,  putting  also  t  =  hJ, 


4-  GO 

If^^dt^l. 


tm 


This  equation  will  give  the  value  of  the  constant  c,  provided  that  the 


value  of  the  integral 


»/  0 


dt 


is  known.  Since  the  definite  integral  is  independent  of  the  variable, 
let  us  multiply  it  b)  a  similar  one,  in  which  y  is  the  variable;  no 
that  we  have 


JW  which  the  ord«»r  of  int»'gratio«  k  indifferent.     If  we  pirt  y  ~  fe, 


METHOD   OF   I-EAST   SQUARES. 


365 


wo  have,  since  t  is  regarded  as  eonstiint  in  tlie  integration  witli  respect 

dy  =  tdz ; 
and  hence 


( f'e-''dty=fdz  re-(i+^')' 

Then,  since  \\c  have,  in  general, 


tdt. 


s: 


-ax» 


^^•^•=^2a' 


the  preceding  equation  gives 


+  z') 


Utau-'^]  =];:, 


i  =  0 


in  which  rr  denotes  the  senii-eircuniference  of  a  circle  whose  radius  is 
unity.     Therefore  we  have 


0 

and  the  equation  (10)  gives 


(11) 

(12) 


Hence,  the  expression  for  (p  (J)  becomes 


V  - 


(13) 


Tlie  constant  h,  according  to  the  relation  h^  =  —  ^k,  must  depend  on 
the  mturc  of  the  observations,  and  will  be  the  same  in  the  case  of 
.xystoms  of  obsevvntions  in  which  the  probability  of  an  error  J  is  the 

same.  Since  //^J*  must  ne(  essarily  be  an  abstract  number,  J  and  v 
must  be  homogfineous. 

J  28.  In  a  given  wries  of  observations,  the  probability  that  for  any 
observation  the  error  will  l>e  witliin  the  limits  —  o  and  +  d  will  be 
expressed  by 

i 

^/ 


-»'A* 


dJ; 


(14) 


and  in  anothv-   $^im  t^  '>bservations,  more  or  less  precise,  the  pro- 


366 


THEORirnCAL   ASTIJOXOMY. 


I)!il)ilitv  tliat  the  error  of  an  observation  is  witliin  tlie  limits  — d'  and 


T  O     WI 


Since 


11 


he 


■  h">ii"- 


lU. 


(15) 


-i-hS 


~_  j'  c-"'^'  dJ  =  ---f  e-"'^'  d  ihJ), 

— «  —liS 

it  appears  that  the  intejjjrals  (14)  and  (15)  are  equal  when  hd  =  h'd'. 
llc'iice,  if  we  put  ^'  =  2/(,  these  integrals  will  be  equal  when  d=^-2d', 
and  an  error  of  a  given  magnitude  in  the  first  series  will  have  the 
same  ju'obability  as  an  error  of  half  that  magnitude  in  the  second 
series.  The  second  series  of  observations  will  therefore  be  twice  as 
accurate  as  the  first  series,  and  the  constant  h  may  be  called  the 
iiwuHnrc  of  prcolx'ion  of  the  observations.  The  greater  the  degree  of 
precision  of  the  observations,  the  greater  will  be  the  value  of  li. 

The  relative  accuracy  of  two  scries  of  observations  may  also  he 
detc'Muined  by  a  comparison  of  the  errors  whic^h  are  committed  with 
e(|ual  facility  in  each  sc.'ries.  If  we  arrange  the  errors  of  the  several 
observations  in  each  series  in  the  order  of  their  absolute  magnitude 
without  reference  to  the  algebraic  si^n,  the  errors  which  occu[)y  the 
same  position  in  reference  to  the  extremes  in  each  case  will  serve  to 
detei'mine  the  relation  sought.  We  select  that,  however,  which  occu- 
pies the  middle  place  in  the  series  of  errors  thus  arranged,  and  since 
the  number  of  errors  which  exceed  this  is  the  same  as  the  number 
of  errors  less  than  this,  if  we  designate  the  error  which  occupies  the 
middle  place  by  /■,  the  probability  that  an  error  is  within  the  limits 
—  r  and  +  r  will  be  equal  to  \.  The  probability  of  an  error  greater 
than  }'  being  the  same  as  the  probability  of  an  error  less  than  /■,  the 
error  /•  is  called  the  jiohahfe  error. 

The  relation  between  r  ami  h  is  easily  determined.     Thus,  we  have 


Mill  give  the  relati 


or,  ptittiug  hJ  —  <, 


-J 


dJ 


5' 


/. 


'V<.-^L 


S144311. 


(16) 


If  we  expand  c"**  into  a  series  of  asc^ct*(iing  powers  of  t,  multiply  by 
dt,  and  integrate  btjtween  the  limits  » *  auJ  T,  we  get 


X 


METHOD   OF    LKAST   8QUAKES. 
.-'^  ,]f  —  Ji 1  T">  _!_  I  1  _l_  I  

„  '      '«  -  ^       3  J     t-  Ti  ^  _  2      '  1  .  2  .  3  ^  M  .  2  .  3  .  4 


3()7 
-&^'.,    (17) 


wliich  converges  rajiidly  when  T  is  small.  To  find  the  value  of  T 
wliieli  corresponds  to  the  value  0.44311  assigned  to  the  integral,  we 
nmipnte  the  value  of  the  series  (17)  for  the  values  0.45,  0.47,  and 
0.49  assigned  to  T,  successively,  and  from  the  results  thus  obtained 
it  is  easily  seen  that  Avhen  the  sum  of  the  terms  of  the  series  is 

0.44311,  we  have 

r==/(r  =  0.47694, 
or 

0.47(594 


h 


(18) 


wliich  determines  the  relation  between  the  probable  error  and  the 
measure  of  precision. 

The  })robability  that  the  error  of  an  observation,  without  regard  to 
sign,  does  not  exceed  nr,  is  expressed  by 


nhr 


(19) 


and  this  integral,  therefore,  indicates  the  ratio  of  the  number  of  obser- 
vations affected  with  an  error  which  does  not  exceed  nr  to  the  whole 
niiniber  of  observations.  Hence,  if  we  assign  different  values  to  n, 
the  integral  (19)  computed  for  the  several  assumed  values  of 

nhr  =  0.47694/i 

will  give  the  relative  number  of  errors  of  a  given  magnitude.  Thus, 
if  we  put  n  =  ^,  we  obtain 

0.2385 

fidiii  which  it  appears  that  in  a  series  of  1000  observations  there 
ought  to  be  264  observations  in  which  the  error  does  not  exceed  A?-. 
It  has  been  found,  in  this  manner,  that  in  tiie  case  of  an  extended 
series  of  observations  the  number  of  errors  of  a  given  magnitude 
a>;signed  by  theory  agrees  very  closely  with  that  actually  given  by 
the  sories  of  ol>servations;  and  hence  we  conclude  that  the  error  com- 
mitted in  extending  the  limits  of  the  summation  in  the  expression  (1) 
to  -  X  and  +  oc,  instead  of  the  finite  limits  which  it  is  pi-esumod 
that  the  actual  errors  cannot  exceed,  is  very  slight,  so  that  the  form 


K-^f 


S^^rr 


ii'-^'v 


308 


tiij:oui:tical  astronomy. 


of  thr  fiinrtion  <f(J)  wliicli  has  boon  derived  may  be  regarded  as  tlmt 
which  he.st  .satisfies  all  the  conditions  of  the  problem. 

129.  The  relative  accuracy  of  different  series  of  observations  may 
also  be  indicated  by  means  of  what  are  cjdled  the  mean  error  and  the 
mean  of  (he  rrrorn  for  each  series,  the  former  beinj;  the  error  whoso 
square  is  equal  to  the  mean  of  the  squares  of  all  the  errors  of  tlio 
series,  and  the  latter  the  mean  of  these  errors  without  reference  to 
their  algebraic  sign. 

Let  £  denote  the  mean  error ;  then,  since  the  number  of  observa- 
tions having  the  error  J  is  m(p  (J),  we  shall  have,  according  to  the 
definition, 

.       J»/Hv(J)  +  J"'m^(J')  +  &c. 


m 


^zlV(J). 


But  the  number  of  possible  erroi's  being  infinite,  the  probability  of 
an  error  J  is  expressed  by  ^  ( J)  dJ,  and  wo  have 


+  Q0 


+  00 


which  gives 


2/i»' 


Hence,  by  means  of  (18),  we  have 


»=  -^r-  =  1.4826r, 
hV2 

r=0.67449e, 


(20) 


(21) 


which  determine  the  relation  between  e  and  r. 

Let  1^  denote  the  mean  of  the  errors,  and  we  shall  have 


which  y-ives 


^0  l/7t^0 


Therefore,  wc  have 


hVn 


)?  =  1.1829r, 
r  =  0.8453j;, 


(22) 
(88) 


for  the  relation  between  r  and  rj. 


METHOD  OF   I-EAST  SQUARES. 


309 


130.  Lot  us  (k'lioto  by  v,  v',  v",  etc.  the  iliHiTonccs  botweoii  any 

assiimeil  vtiliic  of  x  and  the  observetl  values  for  a  given  series  of 

olb'cr  vat  ions,  tlie  number  of  observations  being  denoted  by  in;  then, 

it'  we  put 

Ivv]  =  v'  +  v''  +  v"  +  &c.,  (24) 

and  similarly  in  the  case  of  the  sum  of  any  other  series  of  similar 
terms,  wo  shall  have  for  the  probability  of  the  value  .v„ 


h" 


(25) 


aiul  this  probability  will  be  a  maximum  when  [t't']  is  a  minimum. 
Now  we  have 


v'  =  n' 


v  =  n  —  x„  V  =^  w  —  x„  v"  =  u"  —  .r„  &c. 

11,  n',  a",  &c.  being  the  observed  values  of  x,  and  hence 

\yv]  ==[?(»]  —  2  [;i]  x,  +  mx,^ 

=  Inn]  —  ^^-^  -\-  mix,  —  ^-^  )  . 
?/i  \  m  f 

It  appears,  therefore,  that  [r<;]  will  be  a  minimum  when 


X. 


m ' 


(26) 


and  this  is  a  necessary  consequence  of  the  assumption  that  the  arith- 
metical mean  of  the  observations  gives  the  most  probable  value  of  x, 
according  to  which  the  form  of  the  function  (f  (J)  was  derived.  J'ut 
although  the  arithmetical  mean  is  the  most  probable  value,  yet  wtt 
cannot  aflirm  that  this  is  the  exact  value,  so  long  as  the  number  of 
oh.servations  is  finite.  It  becomes  important,  therefore,  to  determine 
the  degree  of  precision  of  the  arithmetical  mean. 

Lot  .r,|  denote  the  most  probable  value  of  ,7',  for  which  tlie  residuals 
are  r,  v',  v",  &g.,  and  let  .i'^  +  ^  bo  any  other  value  oi'  x.  Then,  since 
we  may  put 

lv2^v-\-v'  +  v"  +  ....==0, 
and 

[vv]  =--  ?«£', 

tho  probability  of  the  value  x^  +  u  will  be  ^^    .  --  , 


P' 


■mh-ij:  +lr) 


Viz'" 


V 


370 


TIIEOnETK'AI.   ASTROXOMY. 


The  probability  that  the  error  of  the  aritlinietical  mean  is  zero  is  in- 
(liaited  by 


,_:  C 


and  w'c  have 


r>' 


I/tt" 


Pe" 


Iii  tlio  case  of  a  siiijfle  observation,  if  /*  denotes  the  probaljility  of 
tlie  error  zero,  and  F'  the  probability  of  the  error  d,  we  have 


P'  =  Pe 


—  hV 


Hence  it  appears  that  if  /(^  denotes  the  nieasnre  of  precision  of  the 
arithmetical  mean  of  m  ol)servations,  the  relation  between  Ii^  and  h, 
the  measure  of  precision  of  an  observation,  is  given  by 


h,^  =:  «i/i' ; 


(27) 


and  if  r^  is  the  probable  error  of  the  arithmetical  mean,  and  e^  its 
mean  error,  avc  have,  according  to  the  equations  (18)  and  (20), 


^•o^ 


r 

£ 


(28) 


V 


m 


These  expressions  determine  the  probable  and  the  mean  error  of  the 
arithmetical  mean  of  a  number  of  observations  when  these  errors  in 
the  ease  of  a  single  observation  are  known. 

131.  The  expressions  for  the  relation  between  the  mean  and  pro- 
bable errors  have  been  derived  for  the  case  of  a  very  large  miniber 
of  observations,  a  number  so  great  that  the  error  of  the  arithmetical 
mean  becomes  equal  to  zero.  In  the  case  of  a  limited  number  of 
observed  values  of  x,  the  residuals  given  by  comparing  the  arith- 
metical mean  with  the  several  observations  will  not,  in  general,  give 
the  true  errors  of  the  observations ;  but  the  greater  the  niunber  of 
observations,  the  nearer  will  these  residuals  approach  the  abr^ohite 
errors.  If  J,  J',  J",  &c.  are  the  actual  errors  of  the  observation.^, 
and  V,  v',  v",  (fcc.  those  which  result  from  the  most  probable  value  ol 
X,  we  shall  have,  denoting  the  arithmetical  mean  by  .%  and  the  true 
valne  by  x^  -f  8, 


A  =  v  —  8, 


A'=.v'~  d, 


J"=.v"  —  d,&c.; 


METHOD   OP   I-EAST  SQUAKES. 


371 


and  hence 


)H£'  =  [J J]  ::^  [iv]  +  mo' 


(20) 


This  equation  will  enable  us  to  detenninc  the  mean  error  of  an  ob- 
servation when  o  is  given;  but,  since  this  is  necessarily  unknown, 
sonic  assumption  in  regard  to  its  value  nuist  be  made.  If  we  assinne 
it  to  be  equal  to  the  mean  error  of  the  arithmetical  mean,  the  re- 
niaiiiing  error  will  be  wholly  insensible,  and  hence  the  equation  (29) 
becomes 


Therefore,  we  sliall  have 
and,  according  to  (21), 


~\m-l' 

Afm  —  l 


r  =  0.6745 


(30) 


(31) 


These  equations  give  the  values  of  the  mean  and  probable  errors  of 
a  single  observation  in  terms  of  th(!  actual  residuals  found  by  com- 
paring the  arithmetical  mean  with  the  several  observed  values. 

The  probable  and  the  mean  error  of  the  arithmetical  mean  will  be 
given  by 


^  m  (m  — 


ly 


r,  =  0.6745 


(32) 


When  the  number  of  observations  is  very  large,  the  probable  error 
of  :a\  observation  and  also  that  of  the  arithmetical  mean  may  be  de- 
termined by  means  of  the  mean  of  the  errors.  If  we  suppose  the 
number  of  positive  errors  to  be  the  same  as  the  number  of  negative 
errors,  the  mean  of  the  errors  without  reference  to  the  algebraic  sign 
gives 


""-^l^' 


ami  hence  we  have,  according  to  (23), 


r  =  0.8453 


For  the  mean  error  of  an  observation  we  have 


e  =  ,l/j,7r  =  1.2533 


m' 


(33) 


(34) 


372 


TIIKORKTIfAL  AHTHONOMY. 


Tl'  the  niiiiil)('r  of  olwcrvfttious  is  very  jj;roat,  the  results  given  liy 
thew!  C(jU!itii)ns  will  aj^reo  with  those  fj;iven  by  ('U))  and  (-il);  but  tor 
any  liinit<'(l  series  of  observed  values,  the  results  obtained  l)y  niean.s 
of  the  mean  error  will  alford  the  j:;reatest  acH'uraey. 

1.'12.  The  relative  aeeuraey  of  two  or  more  observed  values  nf  a 
f|Uanli(y  may  1h'  expressed  by  means  of  what  are  called  their  ir<  It/ldK. 
If  the  ol)servations  are  made  under  })re(!isely  similar  eireumstantrs, 
80  that  there  is  no  reason  for  preferrinj^  one  to  the  other,  they  are  said 
to  have  the  same  wei«i;ht.  The  weiji;ht  must  therefore  depend  on  tiiu 
measure  of  preeision  of  the  observations,  and  hence  on  their  j)robal)le 
errors.  The  unit  of  the  weight  is  entirely  arbitrary,  sinee  only  the 
relative  weights  are  recjuired,  and  if  we  denote  the  weight  by  p,  tlic 
value  of  J)  indicates  the  number  of  observations  of  e((ual  accuracy 
which  must  be  condnned  in  order  that  their  arithmetical  mean  may 
hav(!  the  same  degree  of  preeision  as  the  observation  whose  weight  is 
p.  Hence,  if  the  weight  of  a  single  observation  is  1,  the  arithmetical 
mean  of  in  such  observations  will  have  the  weight  m.  Ijet  the  pro- 
bable error  of  an  observation  of  the  weight  unity  be  denoted  by  /•, 
and  the  probable  error  of  that  whose  weight  is  p'  by  »•';  then,  ac- 
cording to  the  first  of  equations  (28),  we  shall  have 


or 


Vp'' 


-.p'r'\ 


For  the  case  of  an  observation  whose  weight  is  p"  and  whose  pro- 
bable error  is  r",  we  have 


:yV'»: 


--p'r'\ 


from  ^vhieh  it  appears  that  the  iveights  of  tivo  observations  arc  to  each 
other  hiverseli/  as  the  squares  of  their  probable  or  mean  errors,  and, 
aceordinc/  to  (18),  directly  as  the  squares  of  their  measures  of  precision. 
Let  us  now  consider  two  values  of  x,  Avhieh  may  be  designated  by 
x'  and  x",  the  mean  errors  of  these  values  being,  respectively,  e'  and 
e";  then,  if  we  put 


X: 


:  X'   ±  X" 


and  suppose  that  both  x'  and  x"  have  been  derived  from  a  large  num- 
ber m  of  observations  (and  the  same  number  in  each  case),  so  that  the 
residuals  v„  v/,  v'",  &c.  in  the  case  of  x'  and  the  residuals  v„  v/,  (,", 
&c.  in  the  case  of  x"  may  be  regarded  as  the  actual  errors  of  ubser- 


MF.THOI)   OF   LKAfiT  HQUARKH. 


373 


vation,  tlu;  errors  of  the  vulue  of  X^  us  deterniined  from  the  scvorul 
observations,  will  be 

V  ±  v„  v'  ±  !•/,  v"  ±  V,",  Ac. 

Let  the  mean  error  of  A'  be  denoted  by  K;  thou  wo  have 
m£'  =  ^(y±  v,y  =  [I'l']  ±  2[i'iv]  +  [y,i',] ; 

and  since  the  number  of  observed  values  is  supposed  to  be  so  ^reat 
that  the  frecpieney  of  nej!;ative  i)r<>duets  vr,  is  the  same  as  that  of  the 
similar  positive  prodnets,  so  that  [_ni,']    -  0,  this  ecpiation  gives 


or 


i'«  =  £'■'  +    S"\ 


Combining  X  with  a  third  value  x'"  whose  mean  error  is  e'",  the 
moiui  error  of  x'  ±  x"  ±  x'"  will  be  found  in  the  same  manner  to  be 
Kjiial  to  £''^ -'f  e"^+  s'"'';  and  lience  we  have,  for  the  algebraic  sum 
of  any  number  of  separate  values, 


and,  according  to  the  last  of  equations  (21), 

R  ^  Vr'  -h  r''  +  r'"  +  etc., 


(35) 


(3G) 


R  hoing  the  probable  error  of  the  algebraic  sum.     If  the  probable 
errors  of  the  several  values  are  the  same,  we  have 


r  =  r'  ^:^  r"  ^:=  &c, 


and  the  probable  error  of  the  sum  of  m  values  wnll  be  given  by 

R  =z  )Vm. 

IIouco  the  probable  error  of  the  arithmetical  mean  of  m  observed 
values  will  be 

_R  _   ^ 

m       V  m 

which  agrees  nith  the  first  of  equations  (28). 

Let  P  denutf!  thi!  weight  of  the  sum  X,  p'  the  weight  of  x',  and  jj" 
that  of  x" ;  rhen  we  shall  have 


y  _  r'^  +  r'" 


pI  -  ^"  +  ^"' 


^'iu 


^>. 


IMAGE  EVALUATION 
TEST  TARGET  (MT-3) 


l/.A 


fA 


1.0 


I.I 


1^128     |2.5 

■iS    12.2 


us 


C«    ''!2.0 


11:25   i  1.4 


1.6 


'/y 


^>. 


^'J> 


^ 


'V' 


Hiotographic 

Sciences 
Corporation 


23  WEST  MAI'l  SVRECT 

WEBSTER, NY.  USSO 

(716)  873-4503 


\ 


4' 


V> 


4^'^.   <>  ^  ^P?<\ 


^^^<5:\^^ 


'^V 


i? 


374 


THEORETICAL   ASTRONOMY. 


PP 


from  wliich  we  get 

Since  the  unit  of  weight  is  arbitrnry,  we  may  take 


f37) 


P  =7« 


1  .     1   „ 

P  =/7,'&c.; 


and  lience  we   have,  for  the  weight  of  tiie  algebraic  sum  of  any 
nuniljer  of  values, 

or,  whatever  may  be  the  unit  of  weight  adopted, 

1 


P^ 


i  4.  J_  _L  X.  4. 

pf  -r  p>,  T  ^///  -r 


(39) 


In  the  ease  of  a  series  of  observed  values  of  a  quantity,  if  we 
designate  by  r'  the  probable  error  of  a  residual  found  by  comparing 
the  arithmetical  me:in  with  an  observed  value,  by  r  the  probiil)lc 
error  of  the  observation,  by  x^  the  arithmetical  mean,  and  by  n  any 
observed  value,  the  probable  error  of 


according  to  (36),  will  be 


n 


•To  +  V. 


,.'  V  > 


m 


Tq  being  the  probable  error  of  the  arithmetical  mean.   Hence  wc  derive 

'  Wl  —  1 


and  if  we  adopt  the  value 


r'  ^0.8453 


.0  H 


m 


the  expression  for  the  probable  error  of  an  observation  becomes 


r  =  0.8453 


l/m(»i  —  1)' 


(40) 


in  which  [v]  denotes  the  sura  of  the  residuals  regarded  as  positive. 
and  m  the  number  of  observations. 

133.  Let  n,  n',  n",  <&c.  denote  the  ol)8erved  values  of  .r,  and  let/), 
p'y  p"f  &c.  be  their  respective  weights ;  then,  according  to  tiie  deti- 


METHOD  OF   LEAST  SQUARES. 


375 


iicc  wc  derive 


nition  of  the  weight,  the  v:ilue  n  may  be  regarded  as  the  arithmetical 
mean  of  p  olwervatioiis  whose  weight  is  unity,  and  the  same  is  true 
in  tiie  ease  of  n',  n",  &v.  We  thus  resolve  the  given  values  into 
}>  -\-  j>'  +  p"  +  . .  . .  observations  of  the  weight  unity,  tuid  the  arith- 
metiml  mean  of  all  these  gives,  for  the  most  probable  value  of  x, 


pti  +  p'n'  4-  p"'i"  -f  &c. [;jh] 


p-\-p'-\-  V"  +  A^'- 


[;0 


(41) 


The  unit  of  weight  l)eing  entirely  arbitrary,  it  is  evident  that  the 
relation  given  by  this  equation  is  eorreet  as  well  when  the  (juuptities 
;>,  y/,  p" ,  ifec.  are  fractional  as  when  they  are  whole  numbers.  The 
weight  of  .t*u  as  determined  by  (41)  is  expressed  by  the  sum 

and  the  probable  error  of  x^  is  given  by 


'•0  = 


n 


Vp  +  J^^f-V 


(42) 


when  r,  denotes  the  probable  error  of  an  oljservation  whose  weight 
is  unity.  The  value  of  r,  must  be  fouhd  by  mej  ns  of  the  observa- 
tions themselves.  Thus,  there  will  be  p  residuals  e.\])ressed  by 
n  —  a-j,  p'  residuals  expressed  by  »'  —  ,r„,  and  similarly  in  the  ease  of 
n",  n'",  &c.     Hence,  according  to  equation  (31),  we  shall  have 


0.(5745  JtM. 
» ?/i  —  1 


(43) 


in  which  m  denotes  the  number  of  values  to  be  combined,  or  the 
number  of  ((uantitics  n,  n',  11",  &c.  For  the  mean  error  of  j-'y,  we 
have  the  ecpiations 

'  Ml  —  1  ,^, 

V'lp^        \[,n-l)lp] 

If  different  determinations  of  the  (juuntity  x  are  given,  for  which 
the  probable  errors  are  r,  r',  r",  <te.,  the  reei[)rociils  of  the  squares 
of  these  probable  crr'^rs  may  be  taken  as  the  weights  of  the  respective 
values  n,  n',  n",  &c.,  and  we  shall  have 


^0=1 


!L  .L  2-  J.  !!_  _i. 


(45; 


376 


THEORETICAL  ASTRONOMY. 


with  the  probable  error 


I.? 


\ 


*o  — 


JI  +  1+J_4.. 


(46) 


T)ic  mean  errors  may  be  used  in  these  equ'.itiona  instead  of  the  pro- 
bable errors. 

134.  The  results  thus  obtained  for  the  case  of  the  direct  observa- 
tion of  the  quantity  sought,  arc  applicable  to  the  determination  of 
the  (!onditions  for  finding  the  most  probable  values  of  several  un- 
known quantities  when  only  a  certain  function  of  these  quantities  is 
directly  observed.  In  the  actual  application  of  the  formuhe  it  will 
always  be  possible  to  reduce  the  problem  to  the  case  in  which  the 
quantity  observed  is  a  linear  function  of  the  quantities  sought.  Thus, 
let  V  be  the  quantity  observed,  and  c,  ;y,  !^,  &c.  the  unknown  quan- 
tities to  be  determined,  so  that  we  have 

Let  7o,  7^Q,  !^Q,  &c.  be  approximate  values  of  these  quantities  supposed 
to  be  already  known  by  meaus  of  previous  calculation,  and  let  x,  y, 
z,  <fcc.  denote,  respectively,  the  corrections  which  must  be  applied  to 
these  approximate  values  in  order  to  obtain  their  true  values.  Then, 
if  we  suppose  that*  the  previous  apj)roximation  is  so  close  that  the 
Sijuares  and  products  of  the  several  corrections  may  be  neglected,  we 
have 


y--'-%' 


^dV    ,dV    , 


and  thus  the  equation  is  reduced  to  a  linear  form.  Hence,  in  general, 
if  we  denote  by  n  the  difference  between  the  computed  and  the  oi)- 
served  value  of  the  function,  and  similarly  in  the  ease  of  each  obser- 
vation employed,  the  equations  to  be  solved  are  of  the  following 

form : — 

ax   -\-hy  -\-  cz    -f  du   -\-  ew   -\-ft   -f  n  =  0, 
a'x  +  h'y  +  c'z  +  d'u  -f-  e'w  +/'<  +  n'  =  0,  (47) 

a"x  +  b"y  +  c"z  +  d"u  +  e"iv  +f"t  +  n"  =  0, 
Ac.  &c. 

which  may  be  extended  so  a  <  to  include  any  number  of  unknown 
quantities.  If  the  number  oi  equations  is  the  same  as  the  number 
of  unknown  quantities,  the  resulting  values  of  these  will  exactly 
satisfy  the  several  equations ;  but  if  the  number  of  equations  exceeds 
the  number  of  unknown  quantities,  there  will  not  be  any  system  of 


METHOD  OP   LEAST  SQUARES. 


377 


values  for  tliesc  which  will  rctlucc  the  socoikI  nioml)ors  ahsolutely  to 
zero,  aiul  we  can  only  (lotornune  the  values  lor  which  the  errors  for 
the  several  equations,  which  may  he  denoted  hy  r,  v',  v",  etc.,  will  lie 
tliorfe  which  we  may  regard  as  helonging  to  the  most  probable  values 
of  the  unknown  quantities. 

Let  J,  J',  J",  <fec.  be  the  actual  errors  of  the  observed  quantities; 
then  the  probability  that  these  occur  in  the  cise  of  the  observations 
used  in  forming  the  equations  of  condition,  will  be  expressed  by 

P=^(J).V.(J').^(J") 

ami  the  most  probable  values  of  the  unknown  quantities  will  be  those 
which  make  P  a  maximum.  The  form  of  the  function  f  (-/)  has 
been  j.lready  found  to  be 

v(J)  =  -7=e 

and  hence  we  shall  have 


P  = 


hh'h". . .   _  ()fii,t  +  h'i^t  +  it'tyt  4.  Ac.) 


V'-- 


VI  being  the  number  of  observations  or  equations  of  condition.     In 
order  that  P  may  be  a  maximum,  the  value  of 

/t»  J»  +  A"  J''  +  h"'S''  +  &c. 

mui^t  be  a  minimum.     If  the  observations  are  equally  good,  the  ex- 
pression for  P  becomes 


P=~.^e 


■  ft»(Aa  +  A'SH-A"»  +  &c.) 


and  the  condition  of  a  maximum  probability  requires  that 

J2  4-  j'>  +  J"i  4-  &c. 

shall  be  a  minimum.  Hence  it  appears  that  when  the  observations  are 
C(|iially  precise,  the  most  probable  values  of  the  unknown  tpiantities 
are  those  which  render  the  sum  of  the  squares  of  the  residuals  a 
niiniinum,  and  that,  in  general,  if  each  error  is  multiplii!tl  by  its 
measure  of  precision,  the  sum  of  the  squares  of  the  products  thus 
formed  must  be  a  minimum. 

If  we  denote  the  actual  residuals  by  r,  v',  f",  <fec.,  and  regard  the 
observations  as  having  the  same  measure  of  precision,  the  condition 
that  the  sum  of  their  squares  shall  be  a  minimum  gives 

dx    -"'  dy    -"'  d» 


=  0,  &c., 


378 


THEORETICAL  ASTRONOMY, 


or 


-^   ,   ^>  dv         „  dv" 
dx  dx  dx 

dv    ,     .dv'  ^    „dv"    , 
dy  dij  di, 

dv         ,dv'        „dv"    . 
%lz^'  dz-^'    dz-^-' 


=  0, 


&c. 
If  we  tliflll'rontiatc  the  equations 


&c. 


ax    -\-  hy   -\-  cz    -\-  du    +  ^'f    +  /'    +  "■   =  ^'» 
ti'x  +  b'y  +  cz  +  d'n  +  c'w  +  ft  +  n'  =^  v', 
a"x  -f  b''y  +  c"z  +  d"u  +  e"w  +ft  +  n"  =  v", 
&c.  &c. 

with  reHpoct  to  x,  y,  z,  &c.,  successively,  we  obtain 

dv"        „   . 

-J-  =  a  ,  &c. 
dx 


dv 
dx-""' 

dv' 
dx-""' 

dv        , 

-dy-'' 

dv'       ,, 

^"1 

dij 


b",  &c. 


&c. 


&c. 


(fee. 


(•18) 


(49) 


(60) 


Introduoinj;  these  vnhics  into  the  e(iuation>«  (48),  and  substituting  for 
V,  v',  v",  etc.  thoir  vahies  given  by  (49),  we  get 


[aa]  X  +  [nh']  y  -f  [o^]  z  -f-  [ad']  a  -f  [ae]  w  -\-  [«/]  t  -\-  [ow]  --  0, 
[«6]J-  +  [hh]y  +  [6c]  ^  +  [bd]  n  +  [A^'jw  -|-  [6/]<  +  [*"]  "--  0, 
[op]  X  4-  [/><•]  J/  +  [cc]  z  +  [cf^  It  +  [cc]  i<;  +  [r/]  t  +  [c/j]  =.  0, ,  . 
[ad]  X  -(-  [Af/]  y  +  [cf/]  2  +  [dd]  n  +  [f/c]  lo  +  [df]  t  +  [rf/i]  -^  0, 
[««■].(•  +  [Ac].'/  +  ['•''] 2  +  ['^e]M  +  [efi]w  -I-  [f:"n<  +  [ch]  =0, 


(51) 


iu  which 


[aa]  z=aa  -\-  a'a'  -\-  a" a"  -f- 
[o6]  =  a6  -f  a'6'  +  a"b"  + 
[ar]  =  ac  +  a'c'  -f  a"c"  + 
[ii]  .-:  ft6  +  6'6'  +  b"b"  -f 
&c.  <&c. 


(52) 


The  equations  of  condition  are  thus  reduced  to  the  same  number  as 
the  number  of  tlie  unknown  quantities,  and  the  sohition  of  these 
will  give  the  values  for  which  the  sum  of  the  squares  of  the  residuals 
will  he  a  minimum.  These  final  equations  arc  called  nornml  equations. 
When  the  observations  are  not  equally  precise,  in  accordance  with 
the  condition  that  AV  -\-  h'h'^  +  h"h"'^  +  &c.  shall  be  a  miuiinuin, 


METHOD  OF    LEAST  SQUARES. 


379 


cncli  ofumtinn  of  condition  ninst  l)c  nuilti|)li<'(l  hy  tho  mcnsnrc  of 
]tn'('ision  of  the  ohscrviitioii;  or,  since  tho  wci^jht  i.s  propMrtional  to 
tlio  .s(|niire  of  the  niciisun?  of  precision,  each  c(|nation  of  (^onditiim 
innst  he  nnihiplied  hy  the  scpiare  root  of  the  weijjht  of  the  ohscrva- 
tioii,  and  tho  several  e(iuations  of  condition,  heinj;  thns  rcdncjid  to 
till'  same  unit  of  weight,  must  he  comhineil  a.s  indieatejl  l)y  the  e(jua- 
tions  (51). 

\',M).  Ft  will  he  ohserved  that  the  formation  of  the  fii*st  normal 
e(|iiation  is  (jifected  hy  multiplyinji;  eaeli  e(|uation  of  condition  i)y 
the  ('((enicient  of  .c  in  that  e(|Uution  and  then  taking;  tlu^  sum  of  all 
the  cfiuations  thus  formed.  The  seeoiid  normal  ecpiation  is  olttained 
ill  the  same  manner  by  multiplyin<r  hy  the  coetlieient  of  //;  and  thus 
l)y  niultiplyini;  hy  the  coeHicient  of  each  of  the  unknown  (piantities 
the  several  normal  equations  are  fornie<l,  Thes<;  e(]uations  will  j^ene- 
rally  j^ive,  by  elln)ination,  a  system  of  determinate  values  of  tho 
unknown  (juantities  x,  if,  z,  ttc.  But  if  one  of  the  normal  e(|uations 
may  l>e  derived  from  one  of  the  others  by  multiplyinj;  it  by  a  (!on- 
stant,  or  if  one  of  the  e(|uations  may  be  derived  by  a  combination  of 
two  or  more  of  the  remainin}r  e([uations,  the  number  of  distinct  rela- 
tions will  be  less  than  the  nnml)er  of  unknown  quantities,  and  tlu; 
proljjem  will  thus  become  indeterminate.  In  this  case  an  unknown 
fliiantity  may  be  expressed  in  the  form  of  a  linear  function  of  one  or 
more  of  the  other  unknown  quantities.  Thus,  if  the  iiumber  of 
iiulependent  ecjuations  is  one  less  than  tho  number  of  uid<nown 
quantities,  the  final  expressions  for  all  of  these  quantities  except  one, 
will  ho  of  the  form 


X 


+  1% 


y-*'  +  A 


3  =  a"  +  ,rt,  &c.        (53) 


The  coefficients  a,  ,9,  a',  /?',  ttc.  depend  on  the  known  terms  and  co- 
oflicicnts  in  the  normal  equations,  and  if  by  any  means  /  can  be  de- 
termined independently,  the  values  of  x,  y,  s,  &c.  become  determinate. 
It  is  evident,  further,  that  when  two  of  tho  normal  equations  may  be 
rendered  nearly  identical  by  the  introduction  of  a  constant  factor,  the 
problem  becomes  so  nearly  indeterminate  that  in  the  numericjil  uppli- 
cition  the  resulting  values  of  the  unknown  ([uantities  will  be  very 
uncertain,  so  that  it  will  be  necessary  to  expres.s  them  as  in  the  equa- 
tions (53). 

The  indetermination  in  the  case  of  the  normal  e(j[uations  results 
nwessarily  from  a  similarity  in  the  original  efpiations  of  condition, 
and  when  the  problem  becomes  nearly  indeterminate,  the  identity  of 


380 


TIIKORKTICAL   ASTRONOMY. 


the  «Y|uationH  will  ho  closer  in  tlio  nnrinnl  ('(luations  than  in  tlio  Cf|iift- 
tionsot'condition  from  wliicli  tlicy  arc  derived.  Itshonld  he  observctj, 
also,  that  when  we  express  .r,  //,  r,  «<re.  in  t<!rnis  of  /,  an  in  (o.'J),  tlio 
normal  e(iuation  in  /,  which  is  the  one  formed  hv  multiplyint?  hy  tlic 
coeflieiont  of  I  in  each  of  the  e<ination.s  of  <'ondition,  is  not  recinired. 

1.%.  The  elimination  in  the  solution  of  the  equations  (51)  is  most 
eonvenicntly  effected  hy  the  method  of  substitution.  Thus,  the  first 
of  these  equations  gives 


[ab]  [ar^  [ad] 


[rt«]  [«aj  [««] ' 


and  if  we  substitute  this  for  x  in  each  of  the  remaining  normal  equa- 
tions, and  put 


[W]  -  [;;J]  [«6]  =  [66.1], 


[be]  -  [;|*^^  [«c]  ==  [6c.l], 


[fc.n -[;;*][''/]= [i/.i]; 


[CO]  -  ^1^']^  [ac]  =  [eel], 
i'^e]  -^''J^[ae]^^[ce.ll 

[rfd]-[^'^][m/]^[cW.l], 


Led]  -  ^J^^  [ad]  =  [cd.l], 

^/]-[::][o/]=[^/.i]; 

l<ie]-^^[ae]  =  [de.\]. 


(o5) 


[ad] 


(56) 


t^'-^]-Mt"-^^  =  t^-^-iJ' 


[ee]  -  [^^^  [«e]  =  [eel], 


i^n-^^i^^n=^^fn 


^Jfy-^^^=uf-ih 


(57) 


f*^^  -  [««]  t""^  =  f*"-^^' 


[c»] 


M    r      T 
- — -i^  Innl  TT^ 


[aa] 


[o»]  =  [c«.l], 


[rfn]  —  f^'^ka/t]  =  [(?«.!],  [en.]-^[an]  =  [cH.l],    (58) 


we  obtain 


[/'O  -  [^  [«n]  -  Un  '], 


MKTIIOI)   OF    I.KAST   SQUAIIEH. 


381 


[W.l]  //  i-  [he A]  z  -I-  [/../.  1]  ,1  -f  [hr.\]  ir  4-  [hf.\]  '  +  [/m.l]  -.:  0, 
[/.,•.  1]  y  +  [crA]  2  -I  [n/.l]  u  -f-  [r,A]  w  -f-  [r/'.l]  /  -f  [ni.l]  0. 
[Ul]//   i    H.l]z  4-  [</</.!]  H  +  [</r.l]  «.  4-  [.(/".n^  4-  [»//).!]     :  0, 

[b'A]y   -}-  [«-.l]  2    +    [f/r.l]  »    4-    [r,'.l]  Jt>  4-   [,/•.!]  <   4-  [fH.l]  ::  .  0, 


(59; 


Tlu'S(!  (Mjtialioiis  aro  synuiu'triciil,  iiuA  of  llio  same  form  as  the  normal 
(•(Illations,  the  cocrticicMts  being  distingnislied  l>y  writing  tho  nnmt-ral 
1  within  the  hracki-ts. 

Tiio  unUnown  (|uantity  .r  is  thus  eliminated,  an<l  l)y  a.simihir  pro- 
('t'>-  //  may  be  eliminated  from  the  e<|uations  (o{>),  the  resnlting  e(|na- 
tioiis  lieing  nMuU'red  symmetrical  in  form  hv  the  introtluetion  of  the 
numeral  2  within  the  braekets.     Thu.s,  we  put 


^''■^'^-[bhA-]^^'-'^^^^''-^^' 
[/W.I], 


[c(U]-[Jj;^j?|[W.l]=:[«/.2], 


['•/.I] 


[brA] 
IbbA] 


[/>/.!]  =[r/.2]; 


(60) 


[rf(/.l] 


[66.1] 


[6(M]  =  [dfl.2l 


t''^-^]-[6U]t'^^'^^='^'^^'-2^^ 


[Af'n-^l;^w^:i  =  w-^i 


(61) 


^''•^^  ~  [60I  t^'-^^  =^  '^''•2^' 


and  the  equations  become  * 


[f/H.l]-[*jJ-j3[;6,,.i]  =  [rf».2], 


[/»■!] 


[66.1  J 


[6h.1]  =  [>.2], 


(63) 


[cc.2]  z  4-  [of/.2]  u  4-  [pe.2]  If'  4-  [c/.2]  <  -f  [rH.2]  =  0, 
[«^2]z  4-  [fW.2]it  4-  [de.2]iv  +  [({t\2]t  +  [«//i.2]  =  0, 
[fv'.2]  2  4-  [(/('.2]  n  4-  [(V'.2]  ?D  4-  [f/.2]  <  4-  [^'".2]  =  0, 
[r/.2]  2  +  [(i/.2]  u  4-  [^/.2]  t(;  +  [jOr.2]  ^  4"  [>.'2]  -  0. 


(64) 


To  eliminate  z  from  these  equations,  we  put 


t<'''.2]-[-^3[crf.2]  =  [rf<i.3], 


[./.2]-[^f[./.2] 


t*-2]-[^[-21 


:[(ie.3], 


[rf/.3]; 


(65) 


.'{82 


TiiKonirrifAL  astijonomy. 


[r/».'J]       [^[o,.2]  :[r/».:5]. 


t^"--^^[l'2jt''""'^^^^t'"-'^]' 


niul  we  liavc 


[/"■2J  -  [-Jl'^]  [o,.2]  -  [/h.3], 


(67) 


(68) 


A^ain  wo  put,  in  n  f^imilar  manner, 

[//■•••i]  -  [ lil/'Ji']  r///.3]  -  UfAl        [r».3]  -  [;J;  ••!jl|  [</«.3]  =  [c«.4],    (69) 


and  the  equations  are 


[«v.4]  u'  +  ['•r.4]  ^  +  [e«.4]  =  0, 
[f/.4]«;  +  [//.4]<+[//,.4]-:0. 


(70) 


Finally,  to  eliminate  w,  we  put 


[#4]  -  ^  [.^4]  =.  [ff.-yl        [fnA]  -  g;Jj  [CH.4]  =  Un.ni     (71) 


and  the  resulting  equation  is 

[/n.5] 


which  gives 


[//•5] 


(72) 
(73) 


The  value  of  t  thus  found  enables  us  to  derive  that  of  tv  hy  iiwaiis 
of  the  first  of  equations  (70).  The  value  of  w  being  found,  that  of 
11  will  be  obtained  from  the  first  of  equations  (68).  In  like  manm  r, 
the  remaining  unknown  quantities  will  be  determined  by  means  of 
the  equations  (64),  (59),  and  (51).  The  determination  of  the  unknown 
quantities  is  thus  redueed  to  the  solution  of  the  foUowiug  system  of 
e(puitions : 


MKTHOl)   OF    I-EAST  SQl'Anf>». 


-f      1 " 


[""J  L""J 


-f 


[""1 


["''1  „  ,  [""1 

[m]  •'   ^  [na]  '    "^  L""]  "     '^  [.""]  "    "^  [""J  '     "^  t""] 
^  "'"  [///;.  1  ]    "^  [A/>.  1  ]     "^  [hh.  1  ]     "^  [/»/..  1  f  '^  [f>t>.  I  ] 


^+[...2J»+t. 


< 


1( 


i]        '    [r,r>]  [,r.2] 


<+ 


[/"••■•I 
[jtr-^] 


0, 
0, 
0, 
0, 
0, 
0, 


(74) 


tlic  <i»('fliciontH  of  which  will  have  Imjcii  found  in  rho  prtKTss  of  do- 
tirmiiiinj;  the  Hovrnil  luixiiiiiry  «}n!intiti«>s.  It  will  he  (thMTVcd, 
fiirtlHM*,  tliftt  lM)th  in  tho  nornml  (Mjuations  and  in  thoHO  whicli  result 
iit'ti  r  fach  HUccc'SMivcj  oliiuination,  th(^  ('(M'nicicnt.s  which  appear  in  a 
li(iri/iintal  lino,  with  tho  exception  of  the  coenicitMit  involvintr  the 
iilMiliitc  torniH  of  tho  equations  of  condition,  an;  found  also  in  the 
(virrespondinjj  vertical  liiu'.  Tho  form  of  tho  notation  [/>i.lj,  ['"■•1]> 
iVr.  may  bo  symbolized  thus : 


r 0     1       [»/'/0  r       T 


[/?r.  0^  4-  D], 


(75) 


in  wliich  a,  ^9,  y,  donoto  any  three  letters,  and  //  any  numeral. 

'fhe  ('(juations  (74)  are  derived  for  tho  ease  of  six  unknown  (|uan- 
tities,  which  is  the  number  usjuilly  to  \w  determined  in  the  correction 
of  the  elements  of  the  orbit  of  a  heavenly  body;  but  there  will  l)e 
iKHlitViculty  in  oxtendinj;  tho  process  indicated  to  the  ease  of  a  <;reater 
niiiiilu'r  of  (uiknown  (piantities,  except  that  the  nund>er  of  auxiliaries 
syiiihoiizod  generally  by  (75)  increases  very  rapidly  when  the  number 
of  unknown  quantities  is  increased. 

1")7.  In  the  numerical  application  of  the  forraulnc,  when  so  many 
quant Ities  are  to  be  computed,  it  Iwoomes  important  to  be  able  to 
clu'ck  tho  accuracy  of  the  calculation  in  it.s  successive  stages.  First, 
then,  to  prove  the  calculation  of  tho  coefficients  in  the  normal  cpia- 
tiuu.s,  we  put 

o+6-fc4-rf+e  +/  =«, 

a'  +  6'  +  c'  +  rf'  +  e'  +/'  =.  s',  Ac. 

If  we  multiply  each  of  the  sums  thus  formed  by  the  corresponding 
absolute  term  n,  and  take  the  sum  of  all  the  products,  we  have 


384 


TIlKOIliniCAIi   AHTKONO.MY. 


[an]  -1    [/».]  -I-  [rn]  -j    [</»]  -\-  [n,]  +  [/"]        ["»]•  (76j 

Tn  a  similar  iiiiuiiH>r,  iiiiiltiplyiii^  i)y  cadi  of  tli)*  (;()('f!i('i(<ntM  in  the 
original  ('<|iiiitions  of  (Condition,  \vc  tiiid 


[an]  4  [ab]  +  [or]  +  [a,l]  +  [(,.]  +  ["/]  [n>>l 
[ab]  +  [W>]  +  [br]  -f-  [/,,/]  4-  [/>,.]  +  [/./•]  -  -  [H, 
["<  ]  -f  {.be]  -f  ['•'•]  -f-  [rd]  +  [<r]  -f-  [,;;•]  [r.], 
[«r/]  H-  [6r/]  -I-  [r<t]  +  [,/</]  4-  [f/.J  -t    [<//•]        [,/.], 


(77) 


Hence  it  appears  that  if  we  eonipiite  tlie  sums  «,  «',  «",  «'",  &(!.,  and 
form  ['"<J,  [^f"<],  [''"J)  t^'<'.  simuUniiecmsiy  with  the  ealciihitioii  of  the 
eoelli<'ieuts  in  the  normal  e(|uations,  the  e(|niition  (70)  nuist  he  satis- 
fied when  \\w  al)s(thile  terms  of  tht;  normal  e(jiiations  are  correct; 
nnd  the  ('(pnitions  (77)  nnist  Ix;  satisfied  when  the  eoeni<;ient.s  of  the 
unknown  tpiantities  in  the  normal  etpiations  are  eurreet. 

The  accuracy  of  the  calci'.lation  of  the  auxiliary  (piantitics  sym- 
bolized hy  the  ctjuation  {lb)  may  be  proved  in  u  similar  manner. 
Thus,  wc  have 

[6..1]  :=  [H  -  [^Jj   M, 

wliich,  by  means  of  the  first  and  second  of  C(juations  (77),  becomes 
[6«.l]  =  [66] 


lab]       ,  ..  [ab]         -,    ,    ri  n        ["'']  r    n 

[ua]  ■■     ■"       •■    ""       [uu]  '--'''--'       [urtj  •-     J 


or 


[6..1]  -  [66.1]  +  [6r.l]  4-  [6(/.l]  +  [6..1]  +  [hf.l]  ; 


(78) 

and  similarly  wc  derive  the  expressions  for  [e«.l],  ['^/w.l],  &e.  It  is 
obvious,  therefore,  that  the  calculation  of  the  eoelfieients  in  th<'  cmia- 
tions  (59),  (<)4),  (68),  and  (70)  will  be  cheeked  as  in  the  case  of  the 
ooettieients  in  the  normal  equations,  the  auxiliaries  depending  on  s 
beiufj:  determined  as  if  ft,  «',  «",  &c.  were  the  coefficients  of  an  addi- 
tional unknown  quantity  in  the  several  equations  of  condition.  Hence 
we  must  have,  finally, 

\.fo-5]  =  Uf-^l  [«/i.5]  =  [/«.5].  (79) 

If  wc  multiply  each  of  the  equations  (49)  by  its  v,  and  take  the 
sum  of  the  several  products,  we  get 

[av]  X  +  [bv]  y  -f  [cv]  z  -{-  [dv]  u  -\-  [ev]  w  +  [fv]  t  -\-  [vn]  =  [w]. 


METIIOn  OF    I.K.VHT  milAKI-W. 


385 


Hut,  iiM'onlinjj  to  the  (M|iinti»ms  (IS)  tiiid  (50),  we  luivc,  f(»r  the  most 
|ir<ilialih-  values  of  the  unknowii  <|iitiiitities, 

[ar]  --  0,  [In]       0,  [w]  =  0,  &c. ; 

ami  heiiec 

[i'h]  =  [I'l-]-  (««) 

l\'  we  iniiltiply  each  of  the  ecumtioiis  (40)  by  it.s  »,  mid  take  the  mxm 
of  all  the  i»roiluet.s  thiiH  formed,  snlwlitiitiiij;  [rtj]  for  [»•«],  there  re- 

MllltS 

[a„]  X   f-  [A«]  ./  4  [n/]  z  +  [</»]  u  +  [e/,]  «-  +  [//']  <  +  [»»]  -  -  f'-]- 

Siil)stitiiting  in  thin  the  value  of  x  given  by  the  fir.st  norniul  (hjuu- 
tioii,  it  becomes 

[hn.l]  y  +  [c«.l]  2  -f-  [f/«.l]  u  4-  [fH.l]  H'  +  [//i.l]  <  +  [n;t.ij  ---^  [viO. 

in  which 

(81) 


r       n        r      1        [""]  r      t 


Siil)stitutinjr,  further,  for  y  it.s  value  given  by  the  first  of  equations 
(5i)),  and  edutinuing  the  proee^^s  as  in  the  elimination  of  the  unknowii 
ijiiantities  by  suceessive  substitution,  we  obtain  the  following  etjua- 
tiotis: 

[CH.2]  2  +  [dn.2]  u  +  [r».2]  w  +  [/n.2]  t  +  [/i»j.2]  =--  [rr], 
[</«.:i]  «  +  leuM]  w  +  [/«.:5]  <  4  [«"..•}]  --  [n-], 

[e«.4]  w  +  [/«.4]  <  4  [nnA]  =-r  [rr],      (82) 

[hh.G]  =  [i'j>]. 
The  expressions  for  the  auxiliaries  [nH.2],  [?j»..3],  ifec.  arc 

[nu.2]  =  [nn.l]  -  [*J|;JJ  [6».l],  [nn.3]  =  [nH.2]  -  [^^^  [r«.2], 


[»".4]^[«„.3]-[^;;;|][rf«.3]. 


[jjH.o]  =  [nH.4]  —  ^'J"^--'  [cH.4], 


[n«.6]  =  [nn.5]  -  ^'^  [//<.5]. 


(83) 


The  process  here  indieated  may  be  rcatlily  extended  to  the  erne  of  a 
prcater  number  of  unknown  quantities,  and  we  have,  in  general,  when 
//  denotes  the  number  of  unknown  quantities, 


[yv]  =  lnn.fi]. 
2» 


(84) 


386 


THEORETICAL   ASTRONOMY. 


This  o([uation  affords  a  coinplotc  verification  of  the  entire  numerioal 
oalciiliition  involved  in  the  determination  of  the  unknown  (juantitios 
from  liie  original  equations  of  eondition.  Tlius,  after  the  elimination 
has  l)een  completed,  we  substitute  the  resulting  values  of  x,  y,  z,  (V'c. 
in  the  efjuatictus  of  condition,  and  derive  the  corresponding  values 
of  tlie  residuals  r,  r',  v",  <te.  Then,  taking  the  sum  of  the  squares 
of  these,  the  ecpiation  (84)  must  be  satisfied  within  the  limits  of  the 
unavoidable  errors  of  calculation  with  the  logarithmic  tiibles  em- 
ployed. \i  this  eondition  is  satisfied,  it  may  be  inferred  that  the 
entire  (.'aleulation  of  the  values  of  the  unknown  quantities  from  the 
given  equations  of  condition  is  correct. 

1.38.  If  the  values  of  .r,  ?/,  s,  ttc.  thus  found  were  the  absolutely 
exact  values,  the  residuals  r,  v',  v",  &c.  would  be  the  actual  errors 
of  observation.  But  since  the  results  obtained  only  furnish  the  most 
probable  values  of  the  unknown  quantities,  the  final  residuals  may 
differ  slightly  from  the  accidental  errors  of  observation.  Further, 
it  is  evident  that  the  degree  of  precision  with  which  the  several 
unknown  quantities  may  be  determined  by  means  of  the  data  of  the 
problem  may  be  very  different,  so  that  it  is  desirable  to  be  able  to 
determine  the  relative  weights  of  the  different  results. 

It  will  be  observed  that  the  erpressions  for  either  of  the  unknown 
quantities  resulting  from  the  ei  ;inatiou  of  the  others  is  a  linear 
function  of  n,  n',  n",  &c.,  so  that  we  have 


a:  +  a/t  4-  a'n'  +  o."n"  +  a"'«"'  +  ....  =  0, 


(85) 


in  which  the  coeflfieients  a,  a',  a",  &.q.  are  functions  of  the  several 
coefficients  of  the  unknown  quantities  in  the  equations  of  condition. 
If  we  now  suppose  the  equations  of  condition  to  be  reduce  1  to  the 
same  unit  of  weight,  the  mean  error  of  the  several  absolute  terms  of 
the  equations  will  be  the  same,  and  will  be  the  mean  error  of  an 
observation  whose  weight  is  unity.  Thus,  if  e  denotes  the  mean 
error  of  an  observation  of  the  weight  unity,  the  mean  error  of  an 
will  be  as,  that  of  a'n'  will  be  a't',  and  similarly  for  the  other  term.? 
of  (85) ;  and,  according  to  the  equation  (-35),  the  mean  error  of  x' 
will  bo 

e^  =  e  l/a»  +  a"  +  a"»  +  &c.  ^  t  Vj^.  (86) 

Hence  the  weight  of  x  will  be  expressed  by 


v.- 


[»•]' 


(87) 


METHOD  OF   LEAST  SQUARES. 


387 


TiCt  .r,  denote  the  true  'value  of  .r,  namely,  that  which  would  bo 
obtained  it'  the  true  values  of  r,  v',  v",  <S:r.  were  retained  in  the 
second  menihers  of  the  etjuations  of  (jondition  instead  of  puttinj^ 
them  etjual  to  zero;  then  it  is  evident  that  the  expression  for  x,  must 
l)e  that  whieh  would  result  by  substituting^  u  -—  r  in  plaee  of  n  in  the 
llirinulie  for  the  most  probal)lc  value  as  determined  from  the  iictual 
data.     Henec  we  have 

x,-^a(n  —  v)  +  a(n'—  v')  +  ....=:  0, 

and  comparing  this  with  the  expression  (85),  wc  obtain 

X,  =  .c  +  [av]. 

Substituting;  in  this  the  values  of  r,  v',  v",  &c.  given  by  the  etjuations 
(4!»),  there  results 

and  since,  according  to  (85),  x  +  [an]  ==  0,  in  order  to  satisty  this 
oxprcssio!!  for  .r„  we  must  evidently  have 

[a./ J  ^  1,      [a6]  =  0,      [oc]  =  0,      [arf]  =  0,      [oe]  =  0,      [a/]  =  0.  (88) 

Since  the  values  of  the  unknown  quantities  as  determined  by  the 
normal  equations  must  be  the  same  by  whatever  mode  the  elimination 
may  have  been  performed,  let  us  suppose  the  method  of  indctcnninate 
multipliers  to  be  appli<'d  for  the  determination  of  x,  and  let  these 
multipliers  be  designated  by  q,  q',  7",  ttc. ;  then,  the  values  of  these 
factors  are  determined  by  the  condition  that  the  coefficient  of  x  in 
the  final  equation  shall  be  unity,  and  that  the  coefficients  of  the  other 
unknown  (piantities  shall  be  zero.     Hence  we  shall  have 

[f/a]  q  +  [06]  q'  +  [-,c]  q"  +  [«</]  r/"  +  ....  =  1, 
["^]  q  +  UM  </  +  f  bc-\  <i'  +  [/«/]  q'"  +  ....  =  0,  (89) 

iac]  q  +  [/.:]  9'  +  [cc]  </'  -j-  [erf]  9'"  +....  =  0, 
Ac.  &c. 

and  also,  I'etaining  the  residuals  v,  r',  v",  &c.  in  the  formation  of  the 
normal  equations, 

^.  +  [«»] q  f  [6»] 9'  +  i'-n-] (('  +  ...  =  [av] q  +  [6^] 9'  +  [cij 9"  +  . . .  (90; 

Therefore,  since 

X,  +  [o?i]  =  [av], 

and  since  the  first  member  of  this  equation  must  be  identical  with 
the  first  member  of  (90),  we  have 

[ay]  9  +  [6v]  9'  +  [cy]  9"  +  .  . .  =  av  +  oV  +  ol'v"  +  .  .  . , 


888 


THEORETICAL  ASTRONOMY. 


which  gives,  by  expanding  the  several  suras, 

aq    -i-bq'    -\-cq"    -\- <lc{"    +... 
a'q  +  h'(i  +  c'<i'  +  d'q'"  +  . . . 

a:'q  +  b"q'-\-c"q"Jrd",r+--- 
&c.  Ac. 


=  o, 

II 
=  a  , 


(91) 


Multiplying  eacli  of  these  equations  by  its  a,  and  adding  the  pro- 
ducts, the  result  is 

[aa]  q  +  [ai]  ,(  +  [af]  ,('  +  [a  J]  5'"  +  ....  =  [aa], 

which,  by  means  of  the  equations  (88),  reduces  to 


(92) 


Plenco  it  appears  that  the  eliminating  factor  q  is  tlie  reciprocal  of  the 
weight  of  X,  and,  since  the  coefficients  of  q,  q',  q",  &c.  in  tiic  c'(jua- 
tious  (89)  are  the  same  as  those  of  a;,  y,  z,  «fec.  in  the  normal  equa- 
tions, tiiat  if  we  ])ut  [mi]  =  — 1,  [6h]=:0,  [ch]=0,  &v,.,  in  the 
normal  equations,  the  I'esulting  value  of  x  will  be  the  reciprocal  of 
the  weight  of  the  inast  probable  of  this  quantity. 

The  equation  (90)  shows  that  if,  in  the  general  ehmination,  hy 
whatever  method  it  may  have  been  effected,  we  write  [«?'],  [^c],  tV'c. 
instead  of  zero  in  the  second  members  of  the  normal  equations  re- 
sj)cctively,  the  coefficient  of  [ctv]  is  the  reciprocal  of  the  weight  of  .r. 
It  is  obvious  that  it  will  not  be  necessarv  to  know  the  num  'rical 
values  of  [or],  [/>r],  &c.,  since  only  the  coefficient  q  is  required.  Tlie 
most  probable  value  of  .r  is  found  from  (90)  by  the  condition  of  a 
mininuim  of  the  squares  of  the  residuals,  namely,  that 


[ai-]  =  0,        [6y]  =  0,         [c!-]  =  0, 


&c. 


The  j)rocess  here  indicated  for  the  determination  of  the  weight  of 
the  final  value  of  x  is  general,  and  applies  to  the  case  of  any  otlier 
unknown  quantity  provided  that  the  necessary  changes  are  made  ia 
the  notation.  Thus,  the  reciprocal  of  the  weight  of  y  is  deterinin(3(l 
by  writing,  in  the  normal  equations,  — 1  in  place  of  [/>«],  and  putting 
[«»],  [en],  t&'j.  equal  to  zero,  and  completing  the  eliminatioii.  It 
is  also  the  co-^fficient  of  [6i']  in  the  value  of  y  when  the  eliminatioii 
is  effected  with  the  symbols  [av'],  [6i'],  &c.  retained  in  the  second 
members  of  the  normal  equations. 

139.  It  may  be  easily  shown  that  when  the  elimination  is  effectccl 
by  the  method  of  successive  substitution,  as  already  explained,  the 


METHOD   OF   LEAST   SQUARES. 


389 


oocfrioicnt  of  the  unknown  quantity  which  is  miule  the  last  in  the 
oliinination,  in  the  final  equation  tor  its  determination,  is  eijual  to  the 
woigiit  of  the  resulting  value  of  that  (juantity.  Thus,  in  the  eu.se  of 
the  eijuations  for  six  unknown  quantities,  since  the  reciprocal  of  the 
weight  of  the  most  probable  value  of  t  is  the  value  of  t  obtained 
from  the  normal  ecjuations  by  putting//i  =  —  1,  and  an,  bn,  vn,  (te. 
equal  to  zero,  the  equations  (63),  (67),  (69),  and  (71)  show  that  we 
have 

[/«]  -=  [/H.l]  =  [/«.2]  =.  [/«.3]  =:  [/«.4]  ==  [/«.5]  -  -  1, 

and  hence,  according  to  (72),  for  tlie  recipi'ocal  of  the  weight  of  t, 

i>.=  [//-5].  (93) 


[//••5]; 


which  gives 


The  weight  of  t  is  therefore  equal  to  its  coefficient  in  the  final  equa- 
tion which  results  from  the  elimination  of  the  other  unknown  quan- 
tities by  successive  substitution.  Hen(!e,  by  repeating  the  elimination, 
fiiucessively  changing  the  order  of  the  quantities,  so  that  each  of  the 
unknown  quantities  may  have  the  last  place,  the  weights  will  be 
(Ictermiiied  independently,  and  the  agreement  of  the  several  sets  of 
values  for  the  unknown  quantities  will  be  a  proof  of  the  accuracy  of 
the  calculation.  It  is  not  necessary,  however,  to  make  so  many 
repetitions  of  the  elimination,  since,  in  each  case,  the  weights  of  two 
of  the  unknown  ([uantities  will  be  given  by  means  of  the  auxiliaries 
used  in  the  elimination.  Thus,  the  reciprocal  of  the  weight  of  w  is 
ol)tained  by  putting  en  =  —  1,  and  the  other  absolute  terms  of  the 
normal  ecjuations  etpial  to  zero,  and  finding  the  corresponding  value 
of  «'.     This  operation  gives 


[e«.4]  =  -l,         [/».4]  =  0, 
Hence  the  equation  (73)  becomes 


'■*'      -■       [ce.4] 


[ee.^ijf.^ 


and  substituting  this  value  of  t  in  the  last  of  equations  (70),  we  get 


or 


(94) 


W: 


390 


THEORETICAL   AHTUOXOMY. 


/ 


which  gives  the  weight  of  to  in  terms  of  tlie  auxiliary  quantities 
required  in  tlie  determination  of  its  in(j*it  probable  value. 

Jf  the  order  of  elimination  is  now  eompletely  reversed,  so  that  x 
is  made  the  last  in  the  elimination,  the  weights  of  x  and  ti  will  be 
determined  by  the  equations 

_[«a.5]_..  ._  (9») 


[auA] 


IbbAl 


A  third  elimination,  in  whieh  z  and  ?6  are  the  unknown  quantities 
first  determined,  will  give  the  weights  of  these  determinations.  It 
appeal's,  therefore,  that  when  only  four  unknown  quantities  are  to  be 
found,  a  single  repetition  of  the  elimination,  the  order  of  the  quan- 
tities being  completely  reversed,  will  furnisii  at  once  the  weights  of 
the  several  results,  and  check  the  accuracy  of  the  calculation.  When 
there  are  only  two  unknown  quantities,  the  elimination  gives  directly 
the  values  of  these  quantities  and  also  of  their  weights. 

140.  In  the  case  of  three  or  more  unknown  quantities,  the  weights 
of  all  the  results  may  be  determined  without  repeating  the  elimina- 
tion  when  certain  additional  auxiliary  quantities  have  been  found. 
The  weights  of  the  two  whieh  are  first  determined  are  given  in  terms 
of  the  auxiliaries  required  in  the  elimination,  that  of  the  quantity 
which  is  next  found  will  require  the  value  of  an  additional  auxiliary 
quantity,  the  .succeeding  one  will  require  two  additional  auxiliaries, 
and  so  on.  The  equations  (74)  show  that  when  the  substitution  is 
effected  analytically  the  final  value  of  x  Avill  nave  the  denominator 

D  :=  [«a]  [ift.l  ]  lcc.2-]  Idd.S^  lee  A]  [//.5], 

and  this  denominator,  being  the  determinant  formed  from  all  the 
coefficients  in  the  normal  equations,  must  evidently  have  the  same 
value  whatever  may  be  the  order  in  which  the  unknown  quantities 
are  eliminated.  Let  us  now  suppose  that  each  of  the  unkndwn 
quantities  is,  in  succession,  made  the  last  in  the  elimination,  and  let 
the  auxiliaries  in  each  elimination  be  distinguished  from  those  when 
t  is  last  eliminated  by  annexing  the  letter  which  is  the  coeflieient  of 
the  quantity  first  determined;  then  we  shall  have 

D^laa-]  [66.1]  [ce.2]  [rfrf.3]  [ee.4]  [jf^.S] 
=  [aal  [66.1],  [cc.2].  [r/r/.3],  [//.4].  [.e.5] 
=  [«a],  [66.1],  [cc.2]„  [ee.3],  [//.4],  [rW.5] 
=  [««].  [bb.ll  ldd.2l  [ee.il  [ffAl  [.0.5] 
-  [««],  ['■^■1]*  l<fd-^\  [.ee.Sl  [f/Al  [66.0] 
=  IbbX  [cc.l]„  ldd.2l  iee.S]^  [//.4].  [aa.5]. 


METHOD   OF    LEAS'l    SQUARES. 


391 


It  will  be  observed,  however,  that  when  the  order  of  ellminatiou  i.s 
changed,  only  those  auxiliaries  which  involve  the  eoeHieient  of"  the 
quantity  which  is  made  the  last  in  the  changed  order  will  be  changed. 
Hence,  if  we  add  the  distinguisliing  letter  only  to  those  auxiliaries 
•.vhich  have  a  different  value  in  the  new  order,  we  have 

Z>  =  [aa][i6.1]  [cr.2]  [.(hIM]  leeA']  \_fj\rq 
=  [««][W>.1]  [.r.2]  [(W.:?][ir.4]  [<r.0] 
=  [o«][ft6.1]  Dr.2]     [m:i]   L():4]JrW.o] 

:=  [«'0  [C..1]   [d,l.->l  [«■.:]]„  [./J.4],  [i/>.o] 
=  [^6]  Kl],.  ldd.'>l  [ecMl  UJ-^l  ["«-'^]. 

and  from  these  equations  we  obtain 


P„  =  [dilo^ 
;).=:[ec.5]  : 


[ee.4], 
J...4] 

[frA]_ 


■  idd:2] 
[dd.S] 


(96) 


^0.2], 

[c^.2] 

[w.2] 
■  [.cc.lX 


[ii.l], 
[Aft.l] 


[««]. 


by  means  of  which  the  weights  of  the  six  unknown  quantities  may 
be  determined.  The  process  here  indicated  may  be  readily  extended 
to  the  case  of  a  greater  number  of  unknown  quantities.  The  equa- 
tion fjr  y),^  is  identical  with  (94),  the  expression  for  p„  introduces  the 
new  auxiliary  quantity  [./(/'. 4]j,  and  that  for  2>e  introduces  two  new 
luixiliar'"- 

The  exi)ressions  for  the  new  auxiliaries  [.)(y.4],„  [.i(jr.4]^,,  [fr.3]^,  <fcc. 
are  easily  formed  by  observing  that  all  the  auxiliaries  as  far  as  those 
■ivhich  are  designated  by  the  numeral  4  are  not  affected  by  putting  e 
or/ last,  that,  as  far  as  those  which  contain  the  numeral  3,  it  makes 
no  difference  whether  d,  e,  or /is  placed  last,  that  those  distinguishcl 
by  the  numerals  1  and  2  are  not  affected  by  making  c,  d,  e,  or/ the 
last,  and  that  those  designated  by  the  numeral  1  are  unchanged 
unless  a  is  made  the  last.     Thus,  we  obtain 


[//.4].  -  [//3] 


W'^l 


(97) 


.392 


THEORETICAL   ASTRONOMY. 


a'/ 1(1,  alriu, 

[-/•3]e-['if/2]-[;J;;|[(//:2], 


[//•3],=  [//.2]-[f;|[e/.2], 


[//■-il^Lir-S], 


i^m. 


(98) 


I'x^ 


/»» 


[«a.5], 

[«.7.4] 


[i6.4], 


laaAl 


[aa.3] 


[cc.3], 
[«6.3], 


(!t9) 


In  llUo  manner  we  may  derive  the  expre.><sions  for  the  new  auxiliarios 
introihiced  into  the  ec^uations  for  jj^  and  p^.  It  will  bo  exj)edic'nt, 
however,  in  the  actnal  application  of  the  formnhe,  to  eliminate  first 
in  the  order  x,  y,  z,  \i,  tr,  t,  and  the  weights  of  the  resnlts  for  ii,  ir, 
and  t  will  be  obtained  by  means  of  the  first  three  of  equations  (90), 
the  single  additional  auxiliary  required  being  found  by  means  of 
(97).  Then  the  elimination  should  be  performed  in  the  order  t,  w,  u, 
z,  y,  X,  and  we  shall  have 

[«a.4]/p:3] 
[ah.S] 

by  means  of  which  the  weights  of  x,  y,  and  r  will  be  determined. 
The  agreement  of  the  two  sets  of  values  of  the  unknown  quantities 
will  prove  the  accuracy  of  the  numerical  calculation  in  the  process 
of  elimination. 

141.  The  weights  of  the  most  probable  values  of  the  unknown 
quantities  may  also  be  computed  separately  when  certain  auxiliary 
factors  have  been  found,  and  these  factors  arc  those  which  are  intro- 
duced when  the  equations  (74)  are  solved  by  the  method  of  inde- 
terminate multipliers  instead  of  by  successive  substitution.  Tlni!?, 
in  order  to  find  x,  let  the  first  of  these  equations  be  multiplied  by  1, 
the  second  by  A',  the  third  by  A",  the  fourth  by  A'",  and  so  on, 
and  let  the  sum  of  all  these  products  be  taken ;  then  the  equations 
of  condition  for  the  determination  of  the  several  eliminating  factors 
will  be 

[ah-] 


0 


[««] 


^-A', 


[art  J        [oft.lj 

\aa]       [oo.l]  [(•(•.2] 


(100) 


0 


[ae]       [6e.l]  [ce.2]     „      [rfe.3]     ,„ 

[ort]  "^  [WU  I  ^  ^  [cc.2]        '^  [rfrf.3]  ^    ^"  ^  ' 


[art]    '   [66.1]      ^  Ltr.iij        ^   [rfrf.3]        ^  [ee.4]  ' 


METHOD  OF   LEAST  SQUARES. 


393 


To  (Ictorminoy  from  the  last  five  of  e(|uation.s  (74),  let  the  cliniinnting 
iiu'tors*  ho  denoted  hy  />'",  B'  ',  7?'^,  and  /i^,  and  wc  shall  have 


0: 


[/,6.1] 


(101) 


4-Ky:f^"  +  j::;T;^5"'+^'\ 


[66.1J       ['"''•-J  [rW.3]  [<'('.4J 

In  a  similar  manner,  we  ohtain  the  following  ecinations  for  the  de- 
tirinination  of  the  eliminating  factors  necessary  lor  finding  the  values 
of  the  remaining  unknown  quantities: 


0: 

0 


[rrf.2] 
[rrr2] 

t«'.2]'  "^  [f^/.3J  ^     "^  ^   ' 


+  C"", 
[(/..3] 


"       [</r/.3]  +  ^'  ' 
"-[fW.3]+[ee.4]^    +^' 


(102) 


0 


[ee.4] 


+  £' 


The  expressions  for  the  values  of  the  unknown  quantities  will  there- 
fore become 


—  X 


[«a]  "^  [66.1]      ^  [ce.2]       "^  [(W.3i       "^  [ee.4]       "^  [.p]       ' 


•2'-[6U]+[«^I^   +[.W.^>j^    +[^^.4]^   +[#5]^ 


—  z 


[m.2]      [rf».3] 
[w.2]  "^  [(W.3] 


4_  [''"•4]  ^.iv  ,   \. />'->} 


C\ 


-  [rW.3]  +  [ee.4]  ^    +  [.^J.5j  ^ 

[ee.4]  ^  [1.5]  ^  ' 
_  ,  _  [.At  5] 

»r   - 


(103) 


394 


THEonKTKA L   ASTRONOMY. 


The  first  of  tlicsc;  cqiiutioiis  will  f>i\'o  the  recipnx-nl  of  tlio  woijjlit  of 
.r,  wIk'Ii  \v»'  )»ut  [rui]  -  1,  and  tliu  otlu'i"  ahsolute  tiTiiis  ol'  tlie 
iioniiiil  ((iiiiitions  i-cjual  to  zero;  tliu  >sl'coii(1  will  give  the  rcciprociil 
of  the  weight  of //  hy  putting  [//»] -^ — 1,  and  the  other  absohite 
terms  oi"  the  normal  ecjuations  e(iiial  to  zero;  and,  continuing  (he 
process,  finally  the  last  ecpiation  will  give  the  reciprocal  of  the  weight 
of  t  when  we  jjiit  fn  '--  —  1,  and  [''/'],  ['>"],  [t'"]»  ^^^'-  t'M"'i'  t"  '•''•'"• 
It  remains,  therefore,  to  determine  the  particular  values  of  [/>".l], 
[(•/(. 2],  ttc,  and  tiie  exi)ressions  for  the  weights  will  he  complete. 
li'  we  multiply  the  first  of  etpiations  (100)  by  [««],  it  becomes 


[6h.1]  =^  [a»]  A'  +  [bn]. 


104) 


Multii)lying  the  second  of  ecpiations  (100)  by  [an],  and  the  first  of 
(101)  by  [/>/(],  adding  the  products,  and  introducing  the  value  of 
[/>«.!]  just  found,  we  get 

[6c.l] 


[e)i'\  —  [pH.l] 
which  reduces  to 


[^6.1] 


[Ah.I]  -\-  [a/i]  A"  4-  [6«]  B"  =  0, 


ini,-]A"  +  ib„-\.ir  +  [<•«]  =  [c«.2]. 


(105) 


Multiplying  the  third  of  cujuations  (KX))  by  [«u],  the  second  of  (101) 

by  [/>"])  '"'•'  ^''^'  ^^'^^  **^  (1^'-)  kv  [''"]>  "thling  the  products,  and  ro- 
dueing  by  means  of  (104)  and  (lOo),  wc  obtain 

0  ^  Idn]  -  Idn.l}  +  [J^'-J^^  [/>..!] 

which,  by  means  of  the  expressions  for  the  auxiliaries,  is  further  re- 
duced to 

[«»]  A'"  +  [6«]  £'"  +  [ch]  C"  +  [rf»]  =:  [rf/i.3].  (106) 

In  a  similar  manner  we  find,  from  the  remaining  equations  of  (100), 
(101),  and  (102),  the  following  expressions: 


ri07) 


[o»]^l"  +  [in]  -6"+  [ch]  C'  +  [rf/i]  />"+  [en]  =  [eH.4], 

[«»]  ^^  +  [ft»]  iJ>  +  [tvt]  C  +  [rf«]  -O'  +  [e/i]  £"  +  [/«]  =^  [/«-r)] 

The  equations  (104),  (105),  (106),  and  (107),  enable  us  to  find  the 
particular  values  of  [in.l],  [e?i.2],  etc.  requiretl  in  the  expressions  for 
the  reciprocals  of  the  weights.     Thus,  for  the  weight  of  x,  wc  have 

[au]  :=  —  1,         Ibii]  =  [oi]  =  Idii]  =  [ch]  =  [/n]  =  0 ; 


MKTMOD   OF    KKAST   SQUAHES. 

and  those  ('(luatioiis  give 


.305 


.1',  [_n,.2] 


A",  [J«.3] 

If,,.-}-]  =  ~  A\ 


.r, 


Fiir  tli(!  case  of  the  weight  of  //,  we  iiiive 

\b,,-\  -  -  1 ,  \_a„-\  -  [c«]  r^  [f/«]  =-.  [e/i]  =  [/»]  =-  0, 

ami  the  satne  equations  give 

[/>//.!]  --  —  1,  [fH.2]  =  —  li\  \jh,SS-\  =  —  E", 


[>.5]  =  -i:- 


Wc  have,  also,  for  the  weight  of  3, 

[r./i]  -  -1,        [</«.3]=:=-C"',        ^«.4]  =  -C", 

llir  the  weight  of  m, 

[t//.3]-^-l,  [^„.4]=-Z)", 

ibr  llio  weight  of  «', 

[f«.4]  =  -     1, 

and  iinally,  for  the  weight  of  /, 

[/...l]  ---=  -  1. 

Inlrodueing  these  partieular  values  into  the  equations  (103),  the  eor- 
r('s|u»ii(ling  values  of  the  unknown  quantities  are  the  reeiprooals  of 
the  weights  of  their  most  probable  values,  respeetively ;  unci  henee 

we  derive 

1  _  i_     .-r^r     A' A!'     a:"A"'     a'-'A'"     ^M' 
K '~  [««] "''  [6^-1]  "^  V<'-^     W-^  ^  [^'•.4]  +  [jf):oJ' 

iT  -  \hbX\  +  [r..2]  +  [</(/;}]  +  [^■.4]  "^  [#5]  • 
1 


ir-[>w.3]  +  [ec.4]+[./)y5]* 

1  _      1  A'^*^ 

£-[ee.4]+UJ-.5]' 
1  1 


(108) 


i>,       L./J-5]" 

The  eqiuvtions  (103)  and  (108)  will  serve  to  determine  separately 
tlie  value  of  each  unknown  quantity  and  also  that  of  its  weight,  the 


306 


TIIKORKTKAI.   ASTIIOXOMV. 


aiixiliiiry  racfors  A',  A",  II",  tVrc.  Iiaviim  hcon  found  from  the  or|iiu- 
tioiis  ( 1(»0),  ( KH),  and  (102).  If  we  n-vcrso  the  operation  and  w- 
(•oniposc  tlic  (>(|(iutions  (71)  by  nii'ans  of  tin;  t'xprcssions  for  the  un- 
known (|Uantitit's  jrivon  hy  (!<>.'}),  the  conditions  wliirh  iuinicdiatclv 
follfiw  furnish  another  scries  of  (Mjuations  for  the  determination  of  tlu; 
auxiliary  liietors.  The  e(|uations  thus  (h'rived  will  };ive  first  th'  valiici 
of  .1',  /,"',  ("",  D%  and  A"  ;  tlH;n,  those  of  A",  11'",  (''%  Jr-,  and  <n 
on.  They  an;  equally  as  convenient  as  those  already  f^iven,  provided 
that  tlu;  values  of  all  thu  unknown  quantities  urc  required  us  well  m 
their  respective  weij^hts. 

142.  The  fornuuie  already  f;ivcn  for  the  relations  between  the  data 
of  the  problem  and  the  weijjjhts  of  the  most  probable  values  of  tlic 
unknown  quantities,  are  those  which  are  of  the  greatest  practical 
value.  It  will  be  apparent  from  what  has  l)een  derived  that  there 
must  l)c  a  variety  of  methods  which  may  be  applied,  but  that  all  ot" 
these  methods  involve  essentially  the  same  numerical  operations. 
The  peculiar  symmetry  of  the  normal  equations  afltbrds  also  a  variety 
of  ex|)rcssions  applicable  to  the  different  phases  under  which  the 
problem  presents  itself. 

.According  to  the  jroiieral  theory  of  elimination,  the  expression  for 
any  unknown  quantity,  as  determined  irom  the  normal  etpiations, 
ma>'  be  i)ut  in  the  form 


A  A'  A" 


(109) 


in  which  D  is  the  determinant  formed  from  all  the  coeflfieients  of  the 
ludvuown  (piantities  in  the  normal  equations,  and  in  which  ^1,  A',  A'', 
&v.  are  the  partial  determinants  recpiircd  in  the  elimination.  TIiuh, 
A  is  the  determinant  formed  from  the  coefficients  of  all  the  unknown 
quantities  except  .r,  in  all  the  equations  except  the  first;  A"  is  the 
determinant  formed  from  the  coefficients  of  y,  z,  &c.  in  all  the  equa- 
tions except  the  second;  and  the  values  of  A",  A"',  &c.  are  fornied 
in  a  similar  manner.  Now,  since  the  value  of  x  which  results  when 
we  put  [rnt]  ^=:  —  1,  and  the  other  absolute  terms  of  the  normal 
equations  equal  to  zero,  is  the  reciprocal  of  the  w'eight  of  the  mo!-t 
probable  value  of  this  unknown  quantity  as  given  by  (109),  we  have 


P.= 


t> 


(110) 


In  like  manner,  the  expression  for  the  most  probable  value  of  y  will  be 


MKTIIOD   OF    I-F.AST   Stil.A  KF». 


I) 


[/'"] 


ij 


[en']  —  Ac, 


397 
(111) 


/),  />",  11" ,  ttc.  bo'mj;  tli<*  partial  (Ictcniiiiiaiits  foniicd  when  the  co- 
cllii'k'iits  ttt"  y  lire  omitted;  and  I'or  it.i  \veij;;ht  we  have 


P, 


I) 
l}' 


(112) 


The  fnrtiiulfP  for  the  most  prohahle  vahie  of  z  and  for  its  \vei<;ht  arc 
entirely  aiiahtj^oiis  to  tiiose  for  x  and  y,  Mt  that  the  pntcess  hcrt;  indi- 
latcd  may  l>e  extended  to  the  ease  (»f  any  nnnd)er  of  nnknosvn  (pian- 
titics.  Jt  ap[>ears,  therefore,  that  the  \vei;;ht  of  the  most  prohaMe 
value  of  any  nid^nown  (puntity  is  found  hy  dividin^r  the;  (complete 
iltttrminant  of  all  the  eoellieients  hy  the  partial  determinant  formed 
when  wo  omit  the  normal  expiation  eorrespondinjij  particularly  to  this 
unknown  (piantity,  and  when  we  omit  also  the  coefficients  of  this 
(lUiuitity  in  the  remaininf;  normal  ecpiations. 

The  iK'culiar  arranj^ement  of  the  coefficients  in  the  normal  o<pm- 
tious  ahhreviates  somewhat  the  ex})ressions  for  the  several  determi- 
nants.    Thus,  in  the  case  of  three  unknown  quantities,  we  have 

.1  [hh-]  [re]  -  [6c]',  B  =  [rta]  [«•]  —  [f»c]^  C"  -^  [«a]  [Ai]  -  [</A]», 
h    :  [on-]  [bb]  [cc]  +  2[«6]  [6c]  [iic']  ~  [o«]  [bv^  ~  [66]  [«f]'-  [«•]  [-(6]-', 

which  are  all  the  quantities  required  for  finding  simply  the  weights 
of  the  most  probable  values  of  x,  y,  and  z.     The  expression  for  the 

weight  of  z  is 

D 

P,  =-■  -fjrr 

When  there  arc  but  two  unknown  quantities,  we  have 

.4  ==.[66],  J3'=[oa],  Z>  =  [««]  [66]  —  [o6]', 

and  hence 

P» 


[«a]  [66]  —  [ab-y 


[66] 


P»  = 


laa]  [66]  —  [«6]' 


[Ott] 


Wlien  the  number  of  unknown  quantities  is  increased,  the  expressions 
for  the  determinants  necessarily  become  much  more'  complicated,  and 
lieiice  the  convenience  of  other  auxiliary  quantities  is  manifest. 

143.  The  case  has  been  already  alluded  to  in  which  the  determina- 
tion of  the  values  of  the  unknown  (juantities  is  rendered  uncertain 
l)y  the  similarity  of  the  signs  and  coefficients  in  the  normal  equations. 


a»8 


Til K( mi'.TKA I.   ASTK( »NOM V. 


ami  ill  wliifli  tlic  ]M'oI)Iciii  Iiccoiim's  nearly  inilctcriiiiiiatc.  Sniiicfinit's 
it  will  l»c  |»o.ssil»lc  to  ovcrcniiic  the  «lillifiilty  thus  ciinnuitrrcd  hv  a 
siiilalilc  <'haiip'  of  the  cleinciits  to  ho  (Irtcniiiiicd ;  hut,  ^;»'iici'allv,  lid* 
a  coiuplctr  uikI  sutisliictory  .solution,  additional  data  will  l»c  rciiuind. 
It  often  happens,  howi!ver,  that  several  of  the  iinUiiowii  ipiaiititics 
may  Ix;  aeciirately  deteriiiined  from  the  jfiven  ecpiatioiis  when  ilic 
values  of  the  <»thers  are  known,  hut  that  the  <'ertaiiity  of  the  det<  r- 
miiiation  ttf  tlu'  saiue  (piantilies  is  very  (greatly  impaired  when  all 
the  unknown  fiuautities  are  derivecl  simultaneously  from  the  saiiio 
e(|uations.  Let  us  suppose  that  one  of  the  unknown  (juantitien  is, 
from  the  very  nature  of  the  prohlem,  not  suseeptihle  of  an  aeciiratc 
determination  from  the  datji  employed.  The  e(juation.s  will  tlieii 
Iiresent  themselves  in  a  form  approaehin^  that  in  which  tlu'  minilicr 
of  independi'iit  relations  is  one  less  than  th(.'  niimher  of  unknown 
quantities,  so  that  it  will  he  neeessary  to  deterinine  the  other  unkiunvn 
fpiantities  in  tcrniH  of  that  whose  value  is  necessarily  uncertain.  In 
this  case  the  elimination  should  be  so  arraii}j;ed  that  the  ipiaiitity 
which  is  regarded  as  unct.'rtain  is  that  whose  value  would  he  lir»t 
determined.  Then,  if  its  eoelUeient  in  the  Hnal  ecpiatioii,  corro- 
spondinji;  to  (72),  is  very  small,  a  circumstance  which  indit^atcs  at 
once  the  exi.steiioo  of  the  uncertainty  wlien  it  is  not  otherwise  sus- 
pected, the  process  of  elimination  should  not  he  eoin}»leted,  and  the 
auxiliary  fpiantities  should  he  determined  only  as  far  as  those  ro- 
<juircd  in  the  formation  of  the  equation  which  (corresponds  to  the  first 
of  (70).     Thus,  let  t  be  tiie  uncertain  (juantity,  and  we  have 


'"~       [e«.4j'       iee.4]' 


which  must  lie  substituted  for  in  in  the  first  of  equations  (68).  Wr 
thus  obtain  w,  u,  z,  if,  and  x  as  functions  of  t.  If  the  solution  is 
elfeeted  by  means  of  tiie  equations  (103),  let  .?•„,  ?/„,  :„,  tte.  denote  tin- 
values  of  these  unknown  quantities  wiien  we  put  <  — 0;  and  then 
we  shall  have 


Xn=  — 


2/o=- 


«n=  — 


[an']  __  [6h.1] 
[A6.1] 


A' 


[bn.l]  _ 
[A6.1]       [cc.2] 

[(Vi.2]_[rfH.;y 

[cc.2]       [dd'M] 


[cH.2]  ^„  _  [t/».3]  ^,„  _  [g».41  ^^j,^ 


[cc.2] 


[c«.2]  ^,       [f///.3] 


C" 


[dd.^ 
\cnA} 
[ee.5] 


B'" 


[dd.-A] 
[c«.4] 


[ee.4] 


[ee.4] 


,-B", 


(113) 


C", 


METHOD  OF    I-KAST  .SqlAUFX. 

_  _  [>)»:.\]         [r„A] 

[n,A\ 


^o~      [,/,/.:;]       [,r.4J 


/>', 


3J)}» 


OVA) 


and  lionoc 

x~r,+  A't, 


^'•=-[,..4]- 


u  =  I/,  +  7)7,  i«  =^  w,  +  E't. 


(lUj 


As  snoii  as  /  is  (Ictcniiiiu'd  hy  .soino  iiuli'iK'ndciit  (.'<nidi(i<<ii  or  relation, 
ilicsc  (■(jiiatioiis  will  j^ive  the  eorrespondiiij^  values  of./*,  */,  ;,  tV:e.  The 
mean  erntrs  of  .»•„,  y,,,  z^^,  ite.  Iiavinj>;  heeii  determined  l)y  ne^lectinj^  t 
entirely,  it"  we  denote  tlu?  mean  error  of  the  final  ado|>te<l  value  of  t. 
l)v  £„  the  mean  errors  of  the  corre.><[)ondin<5  values  of  the  other 
variahles  will  he  given  by 


^;  --  (O' + '-I'^i^',  s"  --  ('-y + fi'^v.  c = (O''  +  <^"' <^"s'. 


■c=(>:/+^'/>v. 


(e„v+A"i;v. 


(11^) 


These 


in  whieli  ($,),  (e^^),  ite,  denote  the  mean  errors  of  .r„,  »/,„  tt( 
f'urniidie  shf)W,  also,  that  when  one  of  the  variables  is  nej^leeted,  the 
e(|nations  assign  oo  great  a  degree  of  preeision  to  the  results  thus 
olitain(,'d. 

Wlieii  there  arc  two  or  more  unknown  qjiantitios  wliieh  cannot  be 
(leterniined  from  the  data  with  sutlieient  eertainty,  the  prolilcm  must 
lie  treated  in  a  manner  entirely  analogous  to  that  here  indieated;  but, 
since  cases  of  this  kind  will  rarely,  if  ever,  occur,  it  is  not  necessary 
to  pursue  the  subject  further. 

144.  The  weights  which  arc  obtained  for  the  most  probable  values 
of  the  unknown  quantities  enable  us  to  find  the  mean  anil  [)robablc 
errors  of  these  values.  Let  e  denote  the  mean  error  of  an  observa- 
tion whose  weight  is  unity;  then  the  mean  error  of  v  will  be 


(116) 


^P. 


and,  in  like  manner,  the  exprcssioua  for  the  mean  errors  of  y,  2,  u, 
&c.  will  be 

S=-^>  ^=-f-  e„=     '      &C.  (117) 

Vp,  Vp.  ^P„ 

It  remains,  therefore,  to  determine  t*^  ,  value  of  e  by  means  of  the 
final  residuals  obtained  by  comparing  the  observed  values  of  the 
function  with  those  given  by  the  most  probable  values  of  the  va- 


400 


TIIKOUKTICAL    ASTRONOMY. 


rial)l<'s.     rf  tlu'sc  residuals  were  tlic  actuiil  fortuitous  errors  of  oljscr- 
vatioii,  the  nicau  error  of  an  observation  would  l)o 


m  heiiijj;  tlie  number  of  ecjuations  of  eondition.  Tliis  value  is  evi- 
dently an  approximation  to  the  correct  result;  but  since  by  supposinir 
tlie  residuals  v,  r',  v",  iVrc.  to  be  ti.e  actual  errors  of  th((  several  ob- 
served values  of  the  function,  we  ussif;u  too  hij^h  a  de«frei'  of  pre- 
cision to  the  s(!veral  results,  the  true  value  (tf  s  must  necessarilv  iii' 
greater  than  that  j^iven  by  this  ecpiation.  liCt  the  true  values  of  (lie 
unknown  (piautities  be  x  \-  A.r,  //  |-  A//,  2  -|-  as,  ite.,  the  substituti;iii 
of  which  in  the  several  e(|uations  of  condition  wi)uld  j;ive  tlic 
residuals  J,  J',  J",  itc. ;  then  wc;  shall  have 


(('AX-  +  //At/    -I-  c'AZ     f-  </'A(( -\-    V'  n-^  J', 


(118) 


If  we  multiply  each  ol  these  equations  by  its  J,  and  take  the  sum 
of  all  the  products,  we  get 

[a  J]  A.i;  4  [hJ]  A//    I-  [rJ]  A3  +  [(/J]  Atf  -f-  .  .  .  .  +  [rJ]  =:-  [J  J]. 

Hut  if  W(^  multiply  each  of  the  same  e<piatious  by  its  v,  take  tlie  sum 
of  the  i)roduets,  an<l  reduce  by  means  of  (18)  and  (50),  we  obtaiu 

and  Iienco  wc;  derive 

[J J]  ^.  [to]  +  [« J]  A.(;  4-  [/>J]  Ay  +  [cJ]  A3  -f-  [f?J]  Alt  4-  ...  .  (110) 

If  we  form  i!.;'  ii(»rmal  ccpiations  from  (118),  it  will  be  observed  timt 
they  arc  of  the  same  form  as  the  normal  ecpuitions  formetl  from  the 
original  e(|uatious  of  condition,  providcil  that  wo  write  —J  in  phui' 
of  V  ;  and  hence,  according  to  (85),  wc;  have 


We  have  also, 


AX 


aJ  -j-  a  J    -f"  "   -^     "t~ 

[aJ]  =  aJ  +  a' J'  -|-  a" J"  + 

and  the  product  of  these  equations  gives 

[ttJ]  A.'C  :=  oa  J'  +  a'alS'  +  a"tt"J"»  +  .  .  . . 
-f  rto'J J'  4-  rto"JJ"  + 

The  mean  value  of  the  terms  eontaining  JJ',  JJ",  &c.  ia  zero,  and 


;;;  t^r-^ 


COMIUXATION    OV   OHSK'tVATIONS, 


JOl 


8  errors  of  obscr- 


nd  tako  tho  sum 


flir  tlic  mean  values  oi'  J-,  J'-,  J"-,  tVc.  we  must,  in  cai'Ii  case,  writo 
c.     Ili'iiiv  tlio  iiu-an  valiio  of  the  [irotliict  [(f  JJ  A.i'  will  hv 

1111(1  this,  l)y  iiK^aius  of  the  lirst  of  eiiuations  (88),  is  fcrther  nMliicetl  to 

Ill  a  similar  manner,  we  obtain  the  value  r  inr  the  mean  value  of 
each  of  the  products  [/>J]a//,  [cJJa:,  iV:e.  Now,  the  terms  aihlcil  to 
[re]  in  the  second  memher  of  the  e(|uation  (11!>)  are  necessarily  very 
small,  and,  althoii<;h  their  exact  value  cannot  l»c  determined,  we  may 
without  scnsihlc!  error  adopt  the  mean  values  ot"  the  several  terms  as 
lull'  determined,  so  tiuit  the  eipiation  heconies 


(1201 
Therefore,    since 

(121) 


II  heiiifjj    the    numher   of    unknown    <|Uantities. 
|JJ]       iiir,  we  shall  have 

'  III  —  H  ^    HI  —  [I 

l>v  means  of  w-hieh  tlu!  niean  error  of  an  observation  whose  wciirht 
is  unilv  may  be  determined.  When  //  1,  this  eijuatiitn  bi-comes 
identical  with  {'.)()). 

I'or  the  determination  of  the  probable  errors  of  the  linal  values  of 
the  imknown  (luantities,  if  /■  denotes  the  probable  error  of  an  obser- 
vation of  the  weij^ht  unity,  we  have  the  folluwin<f  e(piatlons: — 


r -=:  0.07449  VL'-'L, 


r 


(122) 


»   /\ 


.  ite. 


II").  The  formuhe  which  result  from  the  theory  of  errors  aecordin<i; 
to  uliich  the  method  ol'  lea>{  stpiares  is  derived,  enable  us  to  combine 
the  (lata  furnished  by  observati(»n  so  as  to  overcome,  in  the  ^rcalesl 
il('L;icc  p(i.->ililc,  the  ellect  (»f  those  accidental  crr<ir->  which  no  rcfine- 
niciit  of  tlicory  can  sueeesslully  eliminate.  'I'he  problem  of  the  cur- 
icclion  of  the  a|)proximate  elements  ot"  the  orbit  of  a  heavenly  body 
hv  means  of  ji  series  of  observed  places,  re(piires  the  :ipplication  of 
nearly  all  the  distinct  rcsidts  which  have  been  derive(l.  The  (irst 
:'.|»|iro.\iinat('  elements  of  the  orbit  of  the  body  will  be  determined 
h'eiii  three  or  fliur  observed  places  according  to  the  iiicthods  which 

26 


402 


THEORETICAL   ASTUOXOMY. 


liavo  been  ulrciuly  explained.  In  the  ease  of  a  planet,  if  the  inelina- 
tlon  is  not  very  small,  the  method  of  three  geoeentrie  ])laees  may  be 
employed,  but  it  will,  in  general,  afford  greater  accuracy  and  require 
but  little  additional  labor  to  ba.sc  the  first  determination  on  four 
observed  places,  according  to  the  process  already  illustrated.  In  the 
case  of  a  comet,  the  first  assumption  made  is  that  the  orbit  is  a 
parabola,  and  the  elements  derived  in  accordance  with  this  hypothesis 
may  be  successively  corrected,  until  it  is  appai'cnt  Avhether  it  is  ne- 
cessary to  make  any  further  assumption  in  regard  to  the  value  of  the 
eccentricity.  In  all  cases,  the  approximate  elements  derived  from  a 
few  places  should  be  further  corrected  by  means  of  more  extended 
data  before  any  attempt  is  made  to  obtain  a  more  complete  determi- 
nation of  the  elements.  The  various  methods  by  which  this  pre- 
liminary correction  may  be  effected  have  been  already  sufficiently  de- 
veloped. 

The  fundamental  places  adopted  as  the  basis  of  the  correction  may 
be  single  observed  [)laces  sepai-ated  by  considerable  'ntervals  of  time; 
but  it  will  be  preferable  to  use  places  Avhich  may  be  regarded  as  the 
average  of  a  numbci  of  observations  made  on  the  some  day  or  during 
a  few  days  before  and  after  the  date  of  the  average  or  nonnnl  place. 
The  ephemeris  computed  from  the  approximate  elements  known  may 
be  assumed  to  represent  the  actual  path  so  closely  tl  t,  for  an  interval 
of  a  few  days,  the  difVerence  between  computation  and  observation 
may  be  regarded  as  being  constant,  or  at  least  as  varying  proportion- 
ally to  the  time.  Let  n,  n',  n",  <.tc.  be  the  differences  between  com- 
putation and  observation,  in  the  ease  of  either  s})herical  co-ordinate, 
for  the  dates  t,  f',  t",  &c.,  respectively;  then,  if  the  interval  between 
the  extreme  observations  to  l)e  comI)ined  in  the  formation  of  the 
normal  place  is  not  too  great,  and  if  we  regard  the  observations  as 
c{[ually  j)recise,  the  normal  difference  /;„  between  com})ulation  and 
ol)scrvation  will  I»e  found  by  taking  the  arithmetical  mean  of  the 
several  values  of  n,  and  this  being  a[)plied  with  the  pro|)cr  sign  to 
the  computed  sjiherical  co-ordinate  for  the  date  ^„,  which  is  the  mean 
of  /,  /',  t",  ttc,  will  give  the  corresponding  normal  place.  Jiut  when 
different  weights  p,  p',  p",  &c,  are  assigned  to  the  observations,  the 
value  of  »,,  must  be  found  from 


np  +  n'p'  +  n"p"  + 


"""      i> +/+/'  +  ....     ' 
and  the  weight  of  this  value  will  be  equal  to  the  .sum 

p+p'-]-p"  -{- 


(123) 


roMHIXATION    OF   OHSi:UVATK)N'S. 


403 


The  flate  of  tlie  normal  place  will  bo  ilotorniined  by 


(124) 


If  the  error  of  the  ephonioris  ean  be  considered  as  nearly  constant, 
it  is  not  necessary  to  determine  f^  with  great  |)recision,  since  any  date 
not  differing  much  from  the  average  of  all  may  be  adopted  with  suf- 
ficient accuracy.  It  should  be  olwers'cd  further  that,  in  order  to 
obtain  the  greatest  accuracy  practical)lc,  the  sj)herical  co-ordinates  of 
the  body  for  the  date  ^|,  should  be  computed  directly  from  the  elements, 
so  that  the  resulting  normal  place  may  be  a.s  free  as  possible  from  the 
etfcct  of  neglected  differences  in  the  interpolation  of  the  ephemcris. 

When  tliC  diffc^rences  between  the  computetl  and  the  observed 
places  to  be  combined  for  the  formation  of  a  normal  place  cannot  be 
considered  as  varying  proporti(jnally  to  the  time,  we  may  derive  the 
error  of  the  ephemeris  from  an  e([uation  of  the  form  of  (03)5,  namely, 


£iO 


.4  +  ^r  +  CV», 


the  coefficients  A,  B,  and  C  being  found  from  equations  of  condition 
fonaed  by  means  of  the  several  known  values  of  a^  in  the  case  of 
ouch  of  the  spherical  co-ordinates. 

14().  In  this  way  we  obtain  normal  places  at  convenient  intervals 
throughout  the  entire  period  during  which  the  body  was  observed. 
From  three  or  more  of  these  normal  places,  a  now  system  of  elements 
sliould  be  computed  by  means  of  some  one  of  the  methods  which 
have  already  been  given;  and  these  fundamental  jdaces  being  judi- 
ciiMisly  selected,  the  resulting  elements  will  furnish  a  jiretty  close 
approximation  to  the  truth,  so  that  the  residuals  which  are  Ibund  by 
comparing  them  with  all  the  directly  observed  ])laces  may  be  regarded 
as  indicating  very  nearly  the  actual  errors  of  those  places.  We  may 
then  pi'occod  to  investigate  the  character  of  the  observations  more 
t'tilly.  But  since  the  observations  will  have  been  made  at  many  dif- 
ferent places,  by  different  observers,  Avith  instruments  of  different 
sizes,  and  under  a  variety  of  dissimilar  attendant  circumstances,  it 
may  be  easily  understood  that  the  investigation  will  involve  much 
that  is  vague  and  uncertain.  In  the  theorv  of  errors  which  has  been 
developed  in  this  chapter,  it  has  been  assumed  that  all  constant 
errors  have  been  dulv  eliminated,  and  that  the  onlv  errors  which 
remain  are  those  accidental  errors  which  must  ever  continue  in  a 
greater  or  less  degree  undetermined.     The  greater  the  number  and 


404 


THEORETICA L    AHTHONOMY. 


1 


))crfi'cti()ii  of  the  ohservation.s  oniployod,  the  more  nearly  will  those 
errors  be  doterinined,  iind  the  more  nearly  will  the  law  ot"  their  dis- 
tribution conform  to  that  which  has  been  assumed  as  tlie  basis  of 
the  method  of  least  squares. 

When  all  known  errors  have  been  eliminated,  there  may  yet  remain 
constant  errors,  and  also  other  errors  whose  law  of  distribution  is 
])eculiar,  such  as  may  arise  from  the  idiosyncrasies  of  the  diHereiit 
observers,  from  the  systematic  errors  of  the  adopted  star-places  in 
the  ease  of  differential  observations,  and  from  a  variety  of  f)tiior 
sources;  and  since  the  observations  themselves  furnish  the  only  means 
of  arriving  at  a  knowledge  of  these  errors,  it  becomes  important  to 
discuss  them  in  such  a  manner  that  all  errors  which  may  be  regarded, 
in  a  sense  jnore  or  less  extended,  as  rcfjular  may  be  eliminated. 
AVhen  this  has  been  accomplished,  the  residuals  which  still  retnain 
will  enable  us  to  form  an  estimate  of  the  degree  of  aceui'acy  wiiich 
may  be  attributed  to  the  different  series  of  observations,  in  order  that 
they  may  not  only  be  combined  in  the  most  advantageous  maiiiuT, 
but  that  also  no  refinements  of  calculation  may  be  introduced  which 
are  not  warranted  by  the  quality  of  the  material  to  be  employed. 

The  necessity  of  a  preliminary  calculation  in  which  a  high  degree 
of  accuracy  is  already  obtained,  is  indicated  by  the  fact  that,  however 
conscientious  the  observer  may  be,  his  judgment  is  unconsciously 
warped  by  an  inherent  desire  to  produce  results  harmonizing  well 
among  themselves,  so  that  a  limited  series  of  places  may  agree  to 
such  an  extent  that  the  probable  error  of  an  observation  as  derived 
from  the  relative  discordances  A\;»uld  assign  a  weight  vastly  in  excess 
of  its  true  value.  The  combination,  however,  of  a  large  number  of 
independent  data,  by  exhibiting  at  least  an  approximation  to  the 
absolute  errors  of  the  observations,  will  indicate  nearly  what  the 
measure  of  precision  should  be.  As  soon,  therefore,  as  provisional 
elements  which  nearly  represent  the  entire  series  of  observations  have 
been  found,  an  attemj)t  should  be  made  to  eliminate  all  errors  which 
may  be  accurately  or  approximately  determined.  The  places  of  the 
comparison-stai's  used  in  the  observations  should  be  determined  with 
care  from  the  data  available,  and  should  be  reduced,  by  means  of  tlie 
proper  systematic  corrections,  to  some  standard  system.  The  rodiie- 
tion  of  the  mean  places  of  the  stars  to  apparent  places  should  also  l)e 
made  by  means  of  uniform  constants  of  reduction.  The  observations 
will  tlius  be  uniformly  reduced.  Then  the  perturbations  arising  from 
the  action  of  the  planets  should  be  computed  by  means  of  forniulaj 
whicli  will  be  investigated  in  the  next  chapter,  and  the  observed 


COMBINATION   OF   OBSKRV'ATIONS. 


405 


places  sliould  be  freed  from  these  ]>erturhations  so  as  to  give  the 
places  for  Ji  system  of  osculating  elements  for  a  given  date. 

147.  The  next  step  in  the  process  will  he  to  compare  the  i)ro- 
visional  elements  with  the  entire  series  of  observed  [daces  thus  cor- 
rected; and  in  the  cal(!ulation  of  the  cphemeris  it  will  be  advan- 
tageous to  correct  the  places  of  the  sun  given  by  the  tables  whenever 
observations  are  available  for  that  jnirpose.  Then,  selecting  one  or 
more  epochs  as  the  origin,  if  avc  compute  the  coetttcients  ^1,  B,  C  in 

the  equation 

^0  ^  A  +  Br  +  Cr\  (125) 

in  the  ease  of  each  of  the  spherical  co-ordinates,  by  means  of  eipia- 
tiiiDs  of  condition  formed  from  all  the  observations,  tlie  standard 
ephemeris  may  be  corrected  so  that  it  may  be  regarded  as  rei)re.'<enting 
tile  actual  path  of  the  body  during  the  period  included  by  the  obser- 
vations. AVhen  the  number  of  observations  is  consitlerable,  it  will  be 
more  convenient  to  divide  the  observations  into  groups,  and  use  the 
(lift'erences  between  computation  and  observation  for  provisional 
normal  places  in  the  formation  of  the  e(juations  of  condition  for  the 
(k'termination  of  A,  B,  and  C  It  thus  appears  that  the  corrected 
cphemeris  which  is  so  essential  to  a  determination  of  the  constant 
errors  peculiar  to  each  series  of  observations,  is  obtained  without  first 
having  determined  the  most  probable  system  of  elements.  The  cor- 
rections computed  by  means  of  the  equation  (12o)  being  applied  to 
tlio  several  residuals  of  each  series,  we  obtain  what  may  be  ivgarded 
as  the  actual  errors  of  these  observations.  The  arithmetical  or  pro- 
l)al)le  mean  of  the  corrected  residuals  for  the  series  of  observations 
made  by  each  observer  may  be  regarded  as  the  average  error  of  obser- 
vation for  that  series.  The  mean  of  the  average  errors  of  the  several 
scries  may  be  regarded  as  the  'ictual  constant  error  pertaining  to  all 
tlic  observations,  and  the  comparison  of  this  final  mean  with  the 
moans  found  for  the  ditferent  series,  respectively,  furnishes  the  pro- 
bal)le  value  of  the  constant  errors  due  to  the  peculiarities  of  the 
ol)sorvers;  and  the  constant  correction  thus  found  for  each  observer 
siifiidd  be  ajtplied  to  the  corresponding  residuals  already  obtained. 

In  this  investigation,  if  the  number  of  comparisons  or  the  nund)er 
of  wires  taken  is  known,  relative  weights  proportional  to  the  number 
of  comparisons  may  be  adopted  for  the  combination  of  the  residuals 
for  each  series.  In  this  manner,  observations  which,  on  account  of 
the  jK'culiarities  of  the  observers,  are  in  a  certain  sense  heterogeneous, 
may  be  rendered  homogeneous,  being  reduced  to  a  standard  which 


406 


thp:oketical  astiionomy. 


if^;.y 


}ip])roachc.s  tlie  absolute  in  proportion  as  the  number  and  perfeotioii 
of  the  distinct  series  combined  are  increased.  Whatever  constant 
error  remains  will  be  very  small,  and,  besides,  will  ail'ect  all  places 
alike. 

The  residuals  which  now  remain  must  be  I'cgarded  as  consisting 
of  the  actual  errors  of  observation  and  of  the  error  of  the  adopted 
place  of  the  comparison-star.  IIenc(!  they  will  not  give  the  probable 
error  of  observation,  and  will  not  serve  directly  for  assigning  the 
measures  of  j)recision  of  the  series  of  observations  by  each  observer. 
Let  us,  therefore,  denote  by  e.  the  mean  error  of  the  place  of  the 
comparison-star,  by  s,  tlie  mean  error  of  a  single  com})arison;  then 

will  -.'^-  be  the  mean  error  of  m  comparisons,  and  the  mean  error  of 
Via 

the  resulting  place  of  ti:e  body  will,  according  to  equation  (35),  be 
given  by 


^o^=^  +  C 


m 


(12G) 


The  value  of  Sg,  in  the  case  of  each  series,  will  be  found  by  means  of 
the  residuals  finally  corrected  for  the  constant  errors,  and  the  value 
of  e,  is  supposed  to  be  determined  in  the  formation  of  the  catalogue 
of  star-places  adopted.  Hence  the  actual  mean  error  of  an  observa- 
tion consisting  of  a  single  comparison  will  be 


x/'. 


£,  =  l/m(-V-£/). 


(127) 


The  value  of  e,  for  each  observer  having  been  found  in  accordance 
wuth  this  equation,  the  mean  error  of  an  observation  consisting  of  m 
comparisons  will  be 

Si 


V 


m 


The  mean  error  of  an  observation  whose  weight  is  unity  being  do- 
noted  by  £,  the  weiglit  of  an  observation  based  on  m  comparisons  will 
be 

The  value  of  £  may  be  arbitrarily  assigned,  and  we  may  adopt  for  it 
±:  10"  or  any  other  number  of  seconds  for  which  the  resulting  values 
of  ji  will  be  convenient  numbers. 

When  all  the  observations  are  differential  observations,  and  the  stars 
of  comparison  are  included  in  the  fundamental  list,  if  we  do  not  take 
into  account  the  number  of  comparisons  on  which  each  obs(a'ved 


COMBINATION   OF   OHSERVATIONS. 


407 


id  ill  aeconlanoe 


place  tlopoiuls,  it  will  not  he  iieecssiivy  tc»  consider  s,,  and  wo  may 
tiicii  derive  s,  directly  from  the  residuals  corrected  for  constant  errors. 
Further,  in  the  case  of  meridian  observations,  the  error  which  corre- 
i«|)onds  to  f,  will  l)e  extremely  small,  and  hence  it  is  only  when  these 
are  combined  with  ecinatorial  observations,  or  when  equatorial  obser- 
vations based  on  different  numbers  of  comparisons  are  combined,  tliat 
the  separation  of  the  errors  into  the  two  component  parts  becomes 
necessary  for  a  proper  determination  of  the  relative  M-eij^hts. 

According  to  the  complete  method  here  indicated,  after  having 
oliminateil  as  far  as  possil)le  all  constant  errors,  including  the  correc- 
tions assigned  by  equation  (12o)  to  be  applied  to  the  provisional 
cpliemeris,  we  find  the  value  of  *,  given  by  the  equation 

n=V=[ml•l.]-[H^^  (129) 

ill  which  n  denotes  the  number  of  observatio  m,  m',  hi",  &c.  the 
number  of  comparisons  for  the  respective  observations;  and  c,  v',  v", 
itc.  the  corresponding  residuals.  Then,  by  means  of  equation  (128), 
assuming  a  convenient  number  for  s,  we  compute  the  weight  of  each 
observation.  Thus,  for  exanijile,  let  the  residuals  and  corresponding 
values  of  vi  be  as  follows : — 


AO 

HI 

19 

m 

+  2".0 

5, 

—  1".0 

1, 

—  1  .8 

5, 

+  1  .5 

•5, 

—  0  .4 

10, 

+  4.1 

8, 

—  5  .5 

5, 

0  .0 

5. 

Let  the  mean  error  of  the  place  of  a  comparison-star  be 

e.=  ±2".0; 
then  we  have  n  —  8,  and,  according  to  (129), 

8s,'' =341.78  — 200.0, 

£,  =  ±  4".2. 
Let  us  now  adopt  as  the  unit  of  weight  that  for  which  the  mean  erroi  is 

£  =  ±,r.O; 

then  we  obtain  by  means  of  equation  (128),  for  the  weights  of  tiie 
observations. 


v>hich  gives 


2.5,        2.5, 
respectively. 


5.1,        2.5,        3.6,        2.5,        4.1,        2.5, 


408 


THEORETICAL    ASTRONOMY. 


Ill  this  manner  tho  wciirhts  of  tlie  observations  in  the  series  made 
l)y  each  observer  must  be  determined,  usinj;  throughout  tin;  same 
value  of  £.  Then  the  diti'erenees  between  tlie  jihiees  c()Uij)utod  from 
the  provisional  elements  to  be  eorrected  and  the  observed  plaees  eor- 
re<'ted  for  the  eonstant  error  of  the  observer,  must  be  combined  ac- 
cording to  the  eijuatioiiH  (123)  and  (125),  the  adopted  values  of  p,  y>', 
/>",  ttc.  l)eing  those  found  from  (128).  Tims  will  be  obtained  the 
final  residuals  for  the  formation  of  the  equations  of  condition  from 
which  to  derive  the  most  probable  value  of  the  corrections  to  he 
applied  to  the  elements.  The  relative  weights  of  these  normals  will 
be  intlicated  by  the  sums  formed  by  adding  together  the  weights  of 
the  observations  coud)ined  in  the  formation  of  each  normal,  and  the 
unit  of  weight  will  depend  on  the  adopted  value  of  e.  If  it  be  do- 
sired  to  adopt  a  different  unit  of  weight  in  the  case  of  the  solution 
of  the  ecpiations  of  condition,  such,  for  example,  that  the  weight  of 
an  ecpmtion  of  average  precision  shall  be  unity,  we  may  simply  divide 
the  weights  of  the  normals  by  any  number  -p^  which  will  satisfy  the 
condition  im})osed.  The  mean  error  of  an  observation  whose  weight 
is  unity  will  then  be  given  by 


the  value  of  e  being  that  used  iii  the  determination  of  the  weights  p, 
p',  <S:c. 

148.  The  observations  of  comets  are  liable  to  be  affcv,'L^.d  by  otiior 
errors  in  addition  to  those  which  are  common  to  these  and  to  planet- 
ary observations.  Different  observers  will  fix  upon  different  points 
as  the  proper  point  to  be  oljserved,  and  all  of  these  may  differ  from 
the  actual  position  of  the  centre  of  gravity  of  the  comet;  and  fur- 
ther, on  account  of  changes  in  the  physical  appearance  of  the  coniot, 
the  same  observer  may  on  different  nights  select  different  points. 
These  circumstances  concur  to  vitiate  the  normal  places,  inasmuch  as 
the  resulting  errors,  although  in  a  certain  sense  fox'tuicous,  are  yet 
such  that  the  law  of  their  distribution  is  evidently  different  from 
that  which  is  adopted  as  the  basis  of  the  method  of  least  squares. 
The  impossibility  of  assigning  the  actual  limits  and  the  law  of  dis- 
tribution of  many  errors  of  this  elass,  renders  it  necessary  to  adopt 
empirical  methods,  the  success  of  wliich  will  depend  on  the  discrimi- 
nation of  the  computer. 

If  £„  denotes  the  mean  error  of  an  observation  based  on  m  com- 


COMBINATION  OF  OH8ERVATIONS. 


409 


parisons,  and  s„  the  mean  error  to  be  feared  on  aceount  of  the  pecu- 
liarities of  the  physical  appearance  of  the  comet, 

0       I        c 

will  express  the  mean  error  of  the  residuals;  and  if  n  of  tliese 
residuals  are  combined  in  the  formation  of  a  normal  place,  the  mean 
error  of  the  normal  will  be  given  by 


(130) 


on  m  coni- 


Tlie  value  of  e/  may  be  determined  approximately  from  the  data 
furnished  by  the  observations.  Thus,  if  the  mean  error  of  a  single 
comparison,  for  the  different  observers,  has  been  determined  by  means 
of  tiio  ditferences  between  single  comparisons  and  the  arithmetical 
mean  of  a  considerable  number  of  comparisons,  and  if  the  mean  error 
of  the  place  of  a  comparison-star  has  also  been  determined,  the 
equation  (126)  will  give  the  corresponding  value  of  s^-;  then  the 
actual  ditferences  between  com[nittition  and  observation  obtained  by 
eliminating  the  error  of  the  cphemeris  and  such  constant  errors  as 
may  be  determined,  will  furnish  an  approximate  value  of  e^  by  means 
of  the  formula 

n  " ' 

in  which  h  denotes  the  number  of  observations  combined. 

Sometimes,  also,  in  the  case  of  comets,  in  order  to  detect  the  opera- 
tion of  any  abnormal  force  or  circumstance  producing  ditl'erent  elfects 
in  different  parts  of  the  orbit,  it  may  be  expedient  to  divide  the 
observations  into  two  distinct  groups,  the  first  including  the  observa- 
tions nuule  before  the  time  of  perihelion  passage,  and  the  other 
including  those  subsequent  to  that  epoch. 

149.  The  circumstances  of  the  problem  will  often  suggest  appro- 
priate modifications  of  the  complete  process  of  determining  the  rela- 
tive weights  of  the  observations  to  be  combined,  or  indeed  a  relaxa- 
tion from  the  requirements  of  the  more  rigorous  method.  Thus,  if 
on  account  of  the  number  or  quality  of  the  data  it  is  not  considered 
uece.s.sarv  to  compute  the  relative  weights  with  the  greatest  precision 
attainable,  it  will  suffice,  when  the  discussion  of  the  observations  has 
been  carried  to  an  extent  sufficient  to  make  an  approximate  estimate 
of  tlie  relative  weights,  to  assume,  without  considering  the  number 
of  comparisons,  a  weight  1  for  the  observations  at  one  observatory,  a 


410 


THEORETICAL   ASTHOXOMY. 


/ 


wcifi'lit  'I  for  another  class  of  observations,  \  for  a  third  ehiss,  and  so 
on.  Jl  should  he  cthserved,  also,  that  when  there  are  Imt  few  oliser- 
vations  to  he  eonil)hu'd,  th(!  a|)|)li(ation  of  the  fornmhe  for  the  nicMii 
or  ])rol);d)le  errors  may  hi'  in  a  (h'j:;reo  (idhieious,  the  resuUiii;^  vahics 
of  these  errors  heiii;jj  little  more  than  rude  approximations;  still  (Ik 
mean  or  prohaMe  (;rrors  as  thus  determined  furnish  the  most  reliahic 
means  of  estimatiu}^  the  relative  weights  of  the  observations  nnidc 
by  dilferent  observers,  sinec  otherwise  the  scale  of  weights  woiiM 
dejx'ud  on  the  arbitrary  discretion  of  the  computer.  Further,  in  a 
complete  investigation,  even  when  the  very  greatest  care  has  been 
tiiken  in  the  theoretical  discussion,  on  account  of  independent  known 
circumstances  connected  with  some  particular  observation,  it  may  he 
exj)cdieiit  to  change  arbitrarily  the  weight  assigned  by  theory  to 
certain  of  the  normal  places.  It  may  also  be  advisable  to  rejcot 
entirely  those  observations  whose  weight  is  less  than  a  eei'tain  limit 
which  may  be  regarded  as  the  standard  of  excellence  below  wliieh 
the  observations  siiould  be  rejected;  and  it  will  be  proper  to  reject 
observiitions  which  do  not  aiford  the  data  requisite  for  a  homogeneous 
condiination  with  the  others  according  to  the  princij)les  alrcadv 
explained,  liut  in  all  cases  the  rejection  of  apparently  douI)tfiil 
observations  should  not  be  carried  to  any  considerable  extent  uiilcss 
a  very  large  number  of  good  observations  are  available.  The  nunc 
apparent  discrepancy  between  any  residual  and  the  others  of  a  serie.*, 
is  not  in  itself  sufficient  to  wari'ant  its  rejection  uidess  facts  are 
known  which  would  independently  assign  to  it  a  low  degree  of  i)re- 
cision. 

A  doubtful  observation  will  have  the  greatest  influence  in  vitiating 
the  resulting  normal  place  when  but  a  small  number  of  obscrvetl 
places  are  combined;  and  hence,  since  we  cannot  assume  that  the  law 
of  the  distribution  of  errors,  according  to  which  the  method  of  least 
squares  is  derived,  will  be  comidied  with  in  the  case  of  only  a  lew 
observations,  it  will  not  in  general  be  safe  to  reject  an  observation  pro- 
vided that  it  surpasses  a  limit  which  is  fixed  by  the  adopted  theory 
of  errors.  If  the  number  of  observations  is  so  large  that  the  dis- 
tribution of  the  errors  may  be  assumed  to  conform  to  the  theory 
adopted,  it  will  be  possible  to  assign  a  limit  such  that  a  residual 
which  surpasses  it  may  be  rejected.  Thus,  in  a  series  of  rii  observa- 
tions, according  to  the  expression  (19),  the  number  of  errors  greater 
than  nr  will  be 


m 


nnr 


COMIJINATIOX    OF    OKSKKV.VTIONH. 


411 


ami  when  n  has  a  vuliio  .surh  timt  the  value  of  this  cxpn's.-ioii  is  los.i 
tliaii  ().;'),  tlu;  error  iir  will  liave  a  j.5r('ati'r  jtruhahilily  ajiainst  it  tiian 
fill'  it,  and  lu'iicc  it  may  l)o  rcjeitt'd.  The  exitri'>sion  i'nr  limliii^  tiie 
liiuitiiii;  value  of  n  tiierefore  beeoinos 


Hftf 

-        I     C         at      -r  1  —  . 

/  ff    »^  0  2»l 


Vr. 


(i:U) 


\\\  means  of  this  e<|uatioii  wo  derive  for  {fiveii  values  of  m  the  eor- 
nsponiliu^  values  of  /(/*/•— 0.47094/1,  and  lienee  the  values  of  /;. 
For  couveuient  aj)|)lieatioii,  it  will  he  preferable  to  use  ,-  instead  (jI'  '•, 
ami  if  we  jmt  /i' ^^  U.0744i>yj,  the  limitinj^  error  will  he  i\'z,  and  the 
values  of  //'  eorrespondiug  to  given  values  of  //i  will  be  as  exhibited 
ill  the  following  table. 

TABLE. 


Ill 

6 

1.732 

III 
20 

..' 

HI 

n' 

»i 

n' 

2.241 

55 

2.008 

f»0 

2.17:i 

8 

1.8G3 

25 

2.32(5 

00 

2.(i38 

95 

2.791 

10 

l.OHO 

30 

2.3!)4 

05 

2.()()5 

100 

2.-S07 

12 

2.0:57 

35 

2.450 

70 

2.090 

200 

3.020 

14 

2.100 

40 

2.498 

75 

2.713 

300 

3.143 

Ifi 

2.104 

45 

2.539 

80 

2.734 

400 

3.224 

18 

2.200 

50 

2.576 

85 

2.754 

500 

3.289 

Aecording  to  thi.s  method,  we  first  find  the  moan  error  of  an  obser- 
vation by  means  of  all  the  residuals.  Then,  with  the  value  of  m  as 
till'  argument,  we  take  from  the  table  the  corresponding  value  of  «', 
and  if  one  of  the  residuals  exceeds  the  value  n's  it  must  be  rejectod. 
Airaiii,  finding  a  new  value  of  s  from  the  remaining  m  —  1  residuals, 
and  rejieating  the  operation,  it  -vill  be  seen  whether  another  observa- 
tion slioidd  be  rejected;  and  the  process  may  be  continued  until  a 
limit  is  reached  which  does  not  require  the  further  I'ojeetion  of  ob- 
servations. Thus,  for  example,  in  the  case  of  50  observations  in 
wliieh  the  residuals  —11". 5  and  +7".8  occur,  let  the  sura  of  the 
squares  of  the  residuals  be 

[vv-]  =  320.4. 
Then,  according  to  equation  (30),  we  shall  have 

e  =  ±  2".56. 


412 


THKOUKTICAL    AKTIU)X()MY. 


('orn'spoiidinf;  to  tlic  value  m       50,  ilu'  tiiMe  gives  n'  -'-  2.570,  I'lid 
tlio  litiiiliiiif  value  of  tlie  error  Im-coiucs 

n  t  -  -  (»  .n ; 

and  lience  the  re!<i<liml.s  —  ll".o  and    '  7".8  are  ri'jeeted.     Reecjiu- 
piiting  the  mean  error  of  an  ob.sorvi'.tion,  we  have 


^^±-^^^^r'M. 


47 

In  the  formation  of  a  normal  place,  when  the  mean  error  of  an 
observation  has  been  inferred  from  only  a  small  number  of  observa- 
tions, aecordinj;  to  what  has  been  stated,  it  will  not  be  safe  to  rely 
upon  the  ecjuation  (I'U)  for  the  necessity  of  the  rejection  of  a  doulit- 
ful  observation.  But  if  any  abnormal  inHuence  is  suspected,  or  it' 
any  antecedent  discussion  of  observations  by  the  same  ol)server,  niatle 
under  sin\ilar  circumstances,  seems  to  indicate  that  an  error  of  a  jrivcii 
niagnitud',;  is  hij^hly  improbable,  the  application  of  this  formula  will 
serve  to  confirm  or  remove  the  doubt  already  created.  ]\Iuch  will 
theretbre  depend  on  the  discrimination  of  the  computer,  and  on  his 
knowledge  of  the  various  sources  of  error  whi(!h  ma\  conspire  ((iii- 
tinuously  or  discontinuously  in  the  production  oi'  large  ap[)arent 
errors.  It  is  the  business  of  the  observer  to  indicate  the  circiuii- 
stances  peculiar  to  the  phenomenon  observed,  the  instruments  em- 
ployed, and  the  methods  of  observation;  and  the  discussion  of  tlio 
data  thus  furnished  by  different  observers,  as  far  as  possible  in  ac- 
cordance with  the  strict  requirements  of  the  adopted  theory  of  ernn's, 
will  furnish  results  which  must  be  regarded  as  the  besi  which  can  ho 
derived  from  the  evidence  contributed  by  all  tiie  observations. 

150.  AVhen  the  final  normal  places  have  been  derived,  the  ditHr- 
enccs  between  these  and  the  corre>-j)i»ii  iing  places  computeil  from  the 
provisional  elements  to  be  corrected,  taken  in  the  sense  computiitioii 
minus  observation,  give  the  values  of  w,  n',  n",  etc.  which  are  tho 
absolute  terms  of  the  equations  of  condition.  By  means  of  tlic^^t' 
elements  we  comj)ute  also  the  values  of  the  diif'erential  coefficients  of 
each  of  the  spherical  co-ordinates  with  resj)ect  to  each  of  the  elcment.s 
to  be  corrected.  These  differential  coefficients  give  the  values  of  the 
coefficients  a,  h,  c,  a',  b',  Ac.  in  the  equations  of  condition.  Tlu' 
mode  of  calculating  these  coefficients,  for  different  systems  of  co-or- 
dinates, and  the  mode  of  forming  the  equations  of  condition,  have 
been  fully  developed  in  the  second  chapter.     It  is  of  great  import- 


COUIIKCTION    OF   Till:    KLKMENTS. 


413 


iujci'  that  the  mimcriciil  valiU's  of  tlifso  {'oclliciciUs  wlioiild  he  mw- 
I'lilly  »'li«('k('(l  l»y  (lii'crt  calculiitioii,  assiiiiiiiii^  variations  to  tin-  I'lc- 
iiKiits,  or  l»y  means  ol"  (liircrcnct's  wlu'ii  tiiis  test  can  1h'  succi'ssliiUy 
!i|)|»litM|.  In  assi(rnint;  incicnionts  to  the  clcincnts  in  order  to  clirck 
the  tiirination  oC  tiie  ecjuations,  tliey  slioiild  not  l)i>  so  lar^(!  tliat  the 
iKirleeted  terms  of  tho  second  order  l)ecome  sensilih',  nor  so  .small  that 
tlu'V  do  not  afUtrd  th(!  recinired  certainty  i)v  means  of  the  aurecment 
of  the  corresj)ondini;  variations  of  the  spherical  co-ordinates  as 
uhtained  by  siil)stitution  and  hy  direct  calculation. 

As  soon  as  the  ecjuutions  of  condition  have  been  thus  formed,  wo 
iiiiihiply  each  of  tliem  by  the  scjnare  root  of  its  weight  as  trivcn  by 
till'  adopted  relative  \vei.t!;hts  of  the  nor  iial  places;  and  these  ecpia- 
tioMs  will  thus  be  reduced  to  the  same  weij^ht.  In  general,  the 
iiiiiiierical  values  of  the  coelHcients  will  be  such  that  it  will  be  con- 
venient, althoui^h  not  essential  to  adopt  as  the  unit  of  wei;i;ht  that 
wliich  is  the  average  of  the  weights  of  the  normals,  so  that  the 
iiinnbers  by  which  most  of  the  etpiations  will  be  multiplied  will  not 
(litVcr  much  from  unity.  The  reduction  of  the  equations  to  a  unitt)rm 
measure  of  precision  having  been  etfected,  it  renuiins  to  combine  them 
;ii'c(irding  to  the  method  of  least  s(|nar('s  in  order  to  derive  the  most 
|iroiKil)le  values  of  the  unknown  (piantities,  togc^ther  with  the  relative 
woiglits  of  these  values.  It  should  bo  observed,  however,  that  the 
mimerieal  calculation  in  the  coml)ination  and  solution  of  these  equa- 
tions, and  especially  the  reipiired  agreement  of  some  of  the  checks  of 
the  calculation,  will  be  facilitated  by  having  the  numerical  vidues  of 
tilt'  several  coelfK-ients  not  very  une(iual.  If,  therefore,  the  coeilicient 
'(  of  any  unknown  quantity  x  is  in  each  of  the  equations  numerically 
much  greater  o'"  much  less  than  in  the  case  of  the  other  unknown 
(|iiaiitities,  wc  may  adopt  as  the  corresponding  unknown  quantity  to 
liL'  determined,  not  .r  but  u.r,  v  being  any  entire  or  fractional  number 

siuh  that  the  new  coefficients  -,  — .  &q.  shall  be  made  to  aijreo  in 

inaL:iiitude  with  the  other  coefficients.  The  unknown  (quantity  whoso 
value  will  then  be  derived  by  the  solution  of  the  equations  will  be 
w,  and  the  corresponding  weight  will  be  that  of  v.r.  To  find  the 
weight  of  X  from  that  of  vx,  we  have  the  equation 


V.  =  ^'i^ 


(132) 


In  the  same  manner,  the  coefficient  of  any  other  unknown  quantity 
may  be  changed,  and  the  coefficients  of  all  the  unknown  quantities 
may  thus  be  made  to  agree  in  magnitude  within  moderate  limits,  the 


i 


414 


THKOUiyriCA  I.    ASTIIONOMY. 


ii(lvaiit;ii);o  of  wliidi,  in  tlio  minicrical  solution  of  tlio  ('(inations,  will 
!)('  ajjparcnt  hy  a  coiisidcralioii  of  (lie  mode  of  provinu,-  the  calciihi- 
ti'<n  ol'  11m!  cocllicicnts  in  tin;  normal  ('(|iiatioiis.  It  will  be  fxpciliciit, 
also,  to  lake  for  v  sonic  intc^ial  power  of  10,  or,  when  a  fniftloiial 
valn(>  is  rccpiircd,  the  correspond inijj  decimal.  It  may  be  '.cMiarked, 
(iirllier,  that  tlic  introdnction  of  v  is  frenerally  rcipiireil  only  when 
the  eocllicient  of  one  of  tiu;  nidcnown  (inanlities  is  very  larti;e,  as 
frc((;:ently  happens  in  the  case  of  the  variation  of  the  mean  <lailv 
motion  ft. 

When  the  eoellicicnts  of  some  of  the  i.nknown  (piantities  an: 
extremely  small  in  all  the  e(pi:itions  of  condition  to  be  combined,  an 
approxini.tte  solntion,  and  often  one  which  is  snilieiently  a(('nrate  liir 
the  pnrposes  re(piired,  may  be  obtained  by  first  nejrlectinj'-  tlicsi' 
((nantilics  <'ntirely,  and  afterwards  determinint!;  them  separatelv.  In 
ji'.'Meial,  however,  this  can  only  be  done  when  it  is  (u-rtainly  known 
thai  the  inlhieiiec  of  th(!  ne;;leeted  terms  is  not  of  .sensible  maiinitiidc, 
or  when  at  least  approximate  values  of  these;  terms  are  already  ij:;iv('ii. 
When  wc  adopt  the  approximate  plane  of  the  orbit  as  the  fmnla- 
menfid  plane,  the  eipiations  for  the  Ionf:;itn(le  involve  only  i'our  cK'- 
meiits,  and  the  coellieients  I'  the  variations  of  these  elements  in  tin' 
eipiations  for  the  latitudes  are  always  very  small.  Hence,  lor  an 
approximate  solution,  wc  may  first  solve  the  ecpiations  involviiiii'  tliur 
unknown  tpiantities  as  f  irnished  l)y  the  lont;;itndes,  and  then,  suli~li- 
tntinti;  the  resnltiiiij;  Viilncs  in  the  eipiations  for  the  latitndcs,  they 
will  contain  but  tw(  unknown  (piantities,  namely,  those  which  ^ivc 
tlu!  corrections  io  be  applied  to  ^  and  /. 

lol.  When  the  nuirljer  of  ccpiations  of  condition  is  laruc,  lln' 
eompntation  oi  the  numerical  values  of  tlu;  coeilicieiitH  in  the  iKirMiiil 
e(|nations  will  entail  eonsiderable  labor;  uul  heixv  it  is  dcsiraltlc  li> 
arrange  the  calculation  in  a  convenient  forn:,  applyinjj;  also  the  cli('i'k> 
which  have  been  indicated.  'I'lie  most  convenient  arranu'cmciit  will 
be  to  write  the  loj;'arithin-;  of  tin;  absolute  terms  /(,  /(',  ii'\  Ac.  inn 
horizontal  line,  directly  under  these  the  logarithms  of  the  c(i(Hiri(iits 
It,  d',  d",  iVrc.,  then  the  !oy;aritliinH  of  h,  />',  //',  itc.,  and  so  on.  'flicn 
writini!;,  in  a  correspond inu;  form,  the  values  of  loj>;",  h)ix  n',  A-c.  on  a 
slip  of  paper,  by  brin<j;in<i;  this  successively  over  each  line,  tlicsiinis 
[/(//],  ["'(],  [/>"]i  *Vrc.  will  1)(!  readily  formed.  A^aiii,  writing  en 
aiiolh(!r  lip  of  paper  the  Io<;arithms  of  <(,  a',  a",  tt(\,  and  piacinii 
this  lip  snceessively  over  the  lines  eoutaininir  the  eoellicicnts,  \m' 
derive  the  values  [<"']>  ["'']>  [''*']»  ^^'^'      '^'''*^'  multiplicjition  by  6, '",  <!, 


conuKcrioN  (^r  tmi',  klkmknts. 


415 


Ac.  siicccssivt'ly  is  ctTcctcd  in  ;i  similar  niaiiiicr;  and  llms  will  l»c> 
ilcrivctl  [/>/'],  ['"']>  \/>'l\i  ^^'''m  !'"•'  finally  [/J]  in  the  case  of  six  un- 
known ((uanlitics.  In  ilirminu;  llicsc  sums,  in  tlio  cases  of  sums  of 
|iiisitivi'  and  nciiutivc  »inantitios,  it  is  i-onvcnirnt  as  well  as  condncivo 
(ii  aicuracy  lo  write  the  positive  vali,<'s  in  one  vertical  column  and 
the  ntt;ative  values  in  a  separate  coli  nin,  and  take  the  dillirence  of 
the  sums  of  tlu(  uumhers  in  the  resp.'ctive  columns.  The  proof  of 
the  <alcniation  of  the  eoellicients  of  the  normal  e([nations  is  etlected 
hv  iiitro<lucinj!;  s,  .s',  x",  i^'c;.,  the  alj^ehraie  sums  of  all  the  coellieientrf 
in  the  respective  eipiations  of  comlition,  and  treating;  these  as  the 
('(H'Hicients  of  an  additional  uid<no\vn  ((uantity,  thus  forming'  directly 
the  sums  [."<//],  ['^'•'<'],  [^'•'>']>  \_'''^])  ^^'•'-  Then,  according  to  tin;  eipiations 
(7iJ)  ai\il  (77),  thcM'alues  thus  ("omul  should  ai^reewith  those  ohtaiuecl 
i)v  takini;-  the  corresj)ondinj^  sums  of  the  coellieients  in  the  normal 
oiliialions. 

'1  lie  noruKxl  ecpiations  heiufji;  thus  derived,  tlu"  next  step  in  the 
[Mdccss  is  the  determination  of  tlie  values  of  the  auxiliary  (piautitles 
ii((('>>;M'y  for  the  formation  of  the  eipiations  (7  I).  An  examination 
of  tlie  eipiations  (')!),  ('»')),  tVrc,  hy  means  of  which  these  avixiliarios 
;ii'i'  ilrtcrinined,  will  indica C  at  once  a  c^onvenient  and  ';y>icni,i{it' 
:irr;ni<j,cment  of  the  numericid  cah-ulatio".  Thus,  we  first  write  in  a 
liDii/dntal  line  the  values  of  L-'ff],  ["'f'],  !•'"']»  •  •  •  ["''*]>  l""I)  '""'  *''" 
iciilv  under  them  (he  corresponding^  loj^arithins.  ]ve.\(,  we  write 
iiiiilcr  these,  eonimeiiciiiji;  with   ["/'],  the   values  of  [/>/>  |,  [/>cj,  [/^f/j, 

. .  r/w],  r/;/)l;  then,  add  in  tj;  the  loiijaritlun  of  the   factor  ,      ,    to   the 

l(it;;irillims  of   [''/>],  ['"'],  t'tc,  sucecsively,  we  write;    the  value  of 

,     ,  \<lh^  under  [bh'\.  that  of  ,      ,  r.fc]  under  {hrl,  and  so  on.     Siili- 

tiMclIiiLi'  the  nninhers  in  this  line  from  those  in  (he  line  alxtve,  the 
ililVnvnces  »ive  the  values  -A'  [hh.l],  [fx'A],  .  .  .  [f'-^A  |,  [/^''-l  J,  to  ho 
writlcii  in  the  next  line,  and  tin;  lo}.farithnis  of  these  we  write  directly 
uiuK  r  them.  Tlien  we  write  in  a  1' .ii/ontal  line  the  values  of  [cc], 
[i'/], .  .  [r'.s],  [c/i],  placinj^  [cc]  -.nder  [/k-.I],  and,  havinj;  added   the 

li'irarithin  of  to  the  !oji;arithnis  of  [frc],  [(f<f^,  it<'.  in  succession, 

we  ilcrive,  aceordiiifij  to  the  eipiations  (T);"))  and  ('">S),  the  values  of 
['•'•.I  {i  |'''/.l], .  .  ('••'''.1],  [<'"•!]'  which  are  to  he  placed  under  the  cor- 
ivs[KMidin}j;  quantities  [ir'],  [o(/].  Aw.  Next,  we  suhtraet  from  these, 
ix'speeiivoly,  the  m'oducts 


[hrA] 


U>cAl 


[bnAl 


416 


TIIFORETICAL   ASTRONOMY. 


iuul  thus  derive  the  vjilucs  of  [cc.2],  [cf/.2], . .  [<'«.2],  [c?i.2],  M-hioh 
are  to  he  written  in  the  next  horizontal  line  and  under  theia  tluir 
logarithms.  Then  we  introduce,  in  a  similar  manner,  the  coeiHcionts 
[f/f/],  [(/('], . .  [d)i],  writing  [tW]  under  [cJ.2];  and  from  eaeh  of  these 
in  sueeession  we  subtract  the  products 


[adj 
[rta] 


lad], 


[ad} 
[rtfl] 


[as], 


lad] 
[aa] 


Ian], 


thus  finding  the  values  of  [fW  1],  [f^c.l], . .  [dn .  1].     From  these  we 
subtract  the  products 


Ibd.l] 
[66.1] 


Ibd.l], 


Ihd.l] 
Ihb.l] 


[6e.l], 


[trUJ 
[66.1] 


[6«.l], 


respectively,  which  operation  gives  the  values  of  [(W.2],  [c?c.2], . . . 
[f?/i.2].     From  these  results  we  subtract  the  products 


[cc.2] 


[cf/.2j 
[cc.2] 


[ce.2]. 


[cd2] 
tcc.2] 


[cn.2], 


and  derive  [f?r/..3],  [(:?c.3], .  .  [f/».3]  under  which  we  write  the  cor- 
responding logarithms.     Then  we  introduce  [cc],  [</],  [es],  and  [en], 

writing  [t'c]  under  [t?c.3].     First,  subtracting  p  -,-[ae],  -: — ^  ["/]'•  ■ 
ac  I  I-     J  L     J 

j- — q-t'^^'O?  ^^'^'  S^t  [^'^•1]>  C'/-!]*  ^s.l],  and  [c«.l];  then  subtracting 

from  these  the  products 
[6c.l] 


[66.1] 


[^^.1],  [!u|[Vl]"-[^^^[^"-l]' 


we  obtain  the  values  of  [ee.2],  [e/.2],  [cs.2],  and    [cn.2].     Again, 
subtractinsr 


[cc.2] 


[ce.2], 


[ce.2] 


i^m, 


[cc.2] 


lai.2], 


[ce.2]  "-      -"  [cc.2]  '■■•'•"-"  ■  •  [cc.2] 

we  have  the  values  of  [cc.3],  [c/.3],  [t',s.3],  [cu.3];  and  finally,  sub- 
tracting from  these  the  products 


[rfc.3] 
[(W.3] 


Ide.S], 


^'"'^[rf/.3],..E-^[^-3]. 


Idd.S] 


Idd.S] ' 


we  derive  the  results  for  [fc.4],  [^/.4],  [e.s.4],  and  [c?i.4];  under  which 
the  corresponding  logaritluns  are  to  be  written. 

If  there  are  six  unknown  quantities  to  be  determined,  we  mast 
further  write  in  a  horizontal  line  the  values  of  Ijj],  f fj],  "nd  ]  fn\, 


COREECTION   OF   THE   ELEMENTS. 


417 


;n.2],  which 
r  theia  tlicir 
e  coclKcicnts 
each  of  these 


.'om  these  we 


],  lde.2l . . . 


rntc  the  cor- 
es],  and  ['»(], 

Mm- 

[aaj 
1  subtracting 


1.2].     A 


gain, 


» 


finally,  sub- 

3], 
under  which 

lied,  'vc  nuist 
■'■jI  and  1  ^"1, 


^£ 


placing  Iffl  under  [''J. 4],  and  by  means  of  five  .successive  subtrac- 
tions entirely  analogous  to  what  pi'eeedcs,  and  as  indicated  by  the 
remaining  equations  for  the  auxiliaries,  we  obtain  the  values  of  [//.o], 
[/y.5],  and  [//(.5]. 

The  values  of  [6«.l],  [fs.l],  [c.s\2],  &c.  serve  to  check  the  calcula- 
tion of  the  successive  auxiliary  coetficionts.     Thus  we  must  have 

[66.1]  +  [6^1]  4-  [6f?.l]  +  [6e.l]  +  [6/.1]  =  [6.y.l] 
[6c.l]  +  [c.l]  +  [crf.l]  +  Ice.l]  +  [.f.l]  -^  [r..l],  Ac, 
[«-.2]  +  [crf.2]  +  [C..2]  +  H2]  =--  [C..2], 
[«;.2]  +  [cW.2]  +  [f?6'.2]  +  [rf/;2]  =  lds.2l  &c. 

Hence  it  appears  that  when  the  numerical  calculation  is  arranged  as 

fibove  suggested,  the  auxiliary  coiitaining  .s  must,  in  each  line,  be 
iiiil  to  the  sum  of  all  the  terms  to  the  left  of  it  in  the  same  line 
,    of  those  terms  containing  the  same  distinguishing  numeral  found 

in  a  vertical  column  over  the  last  quantity  at  the  left  of  this  line. 
There  will  yet  remain  only  the  auxiliaries  which  are  derived  from 

[sfi]  and  [?in]  to  be  determined.     These  additional  auxiliaries  will 

be  found  by  means  of  the  formuhe 


[SH.l]  =  [«»] 


[oa] 


[as], 


[8«.3]  =  [.«H,2]-g^[c..2], 


[m.o]  =  [.3,!.4] 


[CC.2] 
[e«.4] 


[..».2]=-[sH.l]-M[6..1], 
[6«.4]  =  [sn.3]  -  [^^^l]  [ds.^l  (133) 


and  tlio 
proc 


[ee.4] 

i.-t'-ns  (81)  and  (83) 


[e«.4],        isn.6]  =  [s«.5] 


[^••5], 


The  arrang>^ment  of  the  numerical 
'  i  Ho  similar  to  that  already  explained. 
Tilt  v^    : '•  uf  [.sn.l],  [s/..2],  &c.  check  the  accuracy  of  the  results 
for  [6n.l],  L   '■^^,  [en. 2],  [f//i.3],  &c.  by  means  of  the  equations 


Ibn.l]  -j-  [CM.I]  +  Idn.'"   ;-  rci.l]  +  [/«.!]  =  [s«.1], 
icn.2]  +  ldn.2i       'c«.2]  +  r/".2]  r_-_z  [.,„.i], 

[rf«.3]  +  ['"'.3]  +  [>.3]  -  r.-.r,], 

[c».4J  +  [>.4]  =  [..«.4], 
[/«.5]  =  Isaf)]. 


(134) 


It  ui'pi.irs  'iirther,  that,  in  the  case  of  six  unknown  quantities,  since 
[fa.'}]      ■  -J .0],  we  have  [,su.G]  =  0. 

Having  thus  determined  the  numerical  values  of  the  auxiliaries 
reqnir  .'d,  wo  arc  prepared  to  form  at  once  the  equations  (74),  by  means 
of  wluch  the  vali;c«  of  the  unknown  quantities  will  be  determined 

27 


418 


THEORETICAL   ASTRONOMY. 


by  successive  siibstitution,  first  finding  t  from  the  last  of  these  equa- 
tions, then  substituting  this  result  '\i\  the  equation  next  to  the  last 
and  thus  deriving  the  value  of  w,  and  so  on  until  all  the  unknown 
quantities  have  been  determined.  It  will  be  observed  that  the  loga- 
rithms of  the  coefficients  of  the  unknown  quantities  in  these  equa- 
tions will  have  been  already  found  in  the  computation  of  the  aux- 
iliaries. 

If  we  add  together  the  several  equations  of  (74),  fii'st  clearing  them 
of  fractions,  we  get 

0  =  iacC]  X  +  ([a61  +  [iS.l])  y  +  ([«c]  +  [6c.l]  +  [cc.2])  z 
+  ({aa  :   •   ^hfl  1]  -f  [cd.2]  +  idd.^)  u 
+  (M  -.      ■    ']  +  [cc.2]  4-  [rfe.8]  +  [ce.4])w  (135) 

+  ([«/]  +  I.    ■-]  +  [C/.2]    +  [r?/.3]  +  [e/.4]  +  Um)t 
+   laiq  +  [6/1.1]  +  [cu.2]  +  [rf/i.3]  +  [e«.4]  +  [//i.5] ; 

and  this  equation  must  be  satisfied  by  the  values  of  x,  y,  z,  &c.  found 
from  (74). 

152.  Example. — The  arrangement  of  the  calculation  in  the  case 
of  any  other  number  of  unknown  quantities  is  precisely  similar;  and 
to  illustrate  the  entire  process  let  us  take  the  following  equation.s, 
each  of  which  is  already  multiplied  by  the  square  root  of  its  weight:— 

0.707.r  +  2.052;/  —  2.3723  —  0.221«  +  6".58  =  0, 
0.471:c  +  IMly  —  1  :\r>z  —  0.085it  +  1  .63  =  0, 
0.2G0.C  +  0.770//  —  0.356^  +  0.483it  —  4  .40  ==  0, 
0.092.C  +  0.343y  +  0.2353  +  0.469it  —  10  .21  =  0, 
0.414.1-  +  1.204;/  —  1.5003  —  0.205k  +  3  .99  =  0, 
0.040.C  +  0.150;/  +  0.1043  f  0.20G«  —   4  .34  =  0. 

First,  we  derive 

[wi]  =  204.313, 
[an]  --=.  +   4.815,  [aa]  =  +  0.971, 
[6w]  =z  +  12.9()1,  [«6]  =  +  2.821,  [66]  =  4-  8.208, 
[cft]  =  —  2.-).C07,  [nc]-^— 3.175,  [6c]  :=  — 9.1G3,  [cc]  =  + 11.028, 
[(/ft]  =  —  10.2   ;,  [arf]  =  — 0.104,  [6rf]  =  — 0.251,  [erf]  =  +  0.938,  [rfd]  :-=  + 0.594, 
[sn]  =-18.139,  [as]  =  +  0.513,  [6,s]  =  +  l.G10,  [cs]  =  —  0.377,  [rf,s]  =+1.177. 

The  values  of  [sn],  {(is\,  [6s],  [cs],  and  [rfe],  found  by  taking  the 
sums  of  the  normal  coefficients,  agree  exactly  with  the  values  ooni- 
puted  directly,  thus  proving  the  calculation  of  these  coefficiente. 
The  normal  equations  are,  therefore, 


NUMERICAL    EXAMPLE. 


419 


0.971X  +  2.821.7  —    8.1753  —  0.104((  +    4.815  =  0, 
2.821.T  4-  8.208//  —    9.1()83  —  0.251  a  +  12.9(U  =.  0, 

—  3.175j;  —  9.1()8//  +  11.0283  +  0.9;38((  —  25.G97  =  0, 

—  0.104.f  —  0.251//  +    0.9383  -(-  0.594(t  —  10.218  =  0. 

It  will  be  observed  tliat  the  coefficients  in  these  equations  are  nu- 
morieally  greater  than  in  the  cquation.s  of  condition;  and  this  will 
generally  be  the  case.  Hence,  if  we  use  logarithms  of  five  decimals 
in  forming  the  normal  equations,  it  will  l)e  expedient  to  use  tables 
of  six  or  seven  decimals  in  the  .solution  of  these  equations. 

Arranging  the  process  of  elim'*  ..aon  in  the  most  convenient  form, 
the  successive  results  are  as  follows : — 


\lb.\]  =  +  0.0123, 


[id]  =  +  0.0562, 
[cc.l]  =  +  0.0463, 
[CC.2J  =  +  0.3895, 


[M.l]  =  +  0.U511, 
[c(/.l]  =  +  0.o97'.t, 
[c<?.2]  ~-^  t-  0.3r,W, 
[fW.lJ  =  4-  0.5S2tt, 
[(/</.2]  =  +  0.3700, 
[liaSi]  =  +  0.0297, 


[fcs.l]  =  +0.1190, 
[cs.\\    ■=  +  1.,'!001, 
[c's.2]  =   1-0.7539, 
[,?«.!]  =  -1-1.2319, 
[(/.S.2]   =  -f  0.7350, 
[,h:A]  =■  -f  0.0297 
[mt.l]  =  180.430, 
[Hrt.2]=   94.552, 
[nn.3]=   23.60S, 
[Hn.4]=   14.698, 


[6».1]=- 
|«M]=- 

[CH.2]  .=  - 
[r/«.l]  =  - 
[,/H.2J  --  - 
[(/;i.3j  -=  — 
isn.l]  --=  - 
[sn.2]  =  - 
[s„.S]=- 
[siiA]  =  0. 


1.0278, 
9.9528, 
5.2507, 
9.V023, 
5.4323, 
0.5143, 
20.6828, 
10.68S9, 
0.5143, 


The  .several  checks  agree  completely,  and  only  the  value  of  [«n.4] 
remains  to  be  proved.     The  equations  (74)  therefore  give 

X  +  2.9052//  —  3.26982  —  0.1071(t  +    4.9588  =  0, 
7j  +  4.50913  +  4.1545?t  —  83.5610  -=  C 
z  +  0.9356it  —  13.4960  =  0, 
?t  — 17.3165  =  0, 
and  from  these  we  get 

«  =  -!- 17".316,       2  =  — 2".705,       7/ =.  +  23".977,       .r  =  —  81".608. 

Tlien  the  equation  (135)  becomes 

0  =  +  0.9710X  +  2.8333//  —  2.72933  +  0.3412ii  —  1.9838, 

which  is  satisfied  by  the  preceding  values  of  the  unknown  quantities. 
If  we  substitute  these  values  of  ;r,  //,  z,  and  u  in  the  equations  of 
condition  already  reduced  to  the  .same  weight  by  multiplioation  by 
tlio  square  roots  of  their  weights,  Ave  obtain  the  residuals 

+  0".67,       — 1".34,       +2".17,        —  2".01,       —  0".40,       —  0".72, 

The  sum  of  the  squares  of  these  gives 

[yv]  =  {_nnA}  =11.672, 

and  the  difference  between  this  result  and  the  value  14.698  already 


420 


TIIEORETICAI>   ASTRONOMY. 


found  is  due  to  the  decimals  neglected  in  the  computation  of  the 
ninnerical  values  of  the  several  auxiliaries.  The  sum  of  all  the 
ccjuations  of  condition  gives  generally 


[a]x-  +  [6]y  +  [c]3  +  [d]  u  +  . . . .  +  [H]  -  M, 


(136) 


which  may  be  used  to  check  the  substitution  of  the  numerical  values 
in  the  determination  of  v,  v',  &c.  Thus,  we  have,  for  the  values 
here  given. 


1.984.r  +  5M6y  —  5.6103  +  0.647«  —  6.75  =  [v] 


l."63. 


It  remains  yet  to  determine  the  relative  weights  of  the  resulting 
values  of  the  unknown  quantities.  For  this  purpose  we  may  apply 
any  of  the  various  methods  already  given.  The  weights  of  u  and  ; 
may  be  found  directly  from  the  auxiliaries  whose  values  have  been 
computed.     Thus,  we  have 


p^  =  Idd.S-]  =  0.0297, 


P'-m^A^''-^^=^'-'''^' 


If  we  now  completely  revei'se  the  order  of  elimination  from  the 
normal  equations,  and  determine  x  first,  we  obtain  the  values 


and  also 

X  =  —  82."750, 


[ii.2]  =  +  0.0425, 
[ort.3]  =  +  0.00056, 


[a«.2]  =:  +  0.0033, 
[»m.4]  =  14.665, 


2/  =  +  24."365,       z  =  —  2."699,      ti  =  -{- 1:."272. 


The  small  dift'erences  between  these  results  and  those  obtained  by  the 
first  elimination  arise  from  the  decimals  neglected.  This  second 
elimination  furnishes  at  once  the  weights  of  x  and  y,  namely, 


p^  =  [«a.3]  =  0.00050,         p 


[««-3] 
tart.2] 


[66.2]  =  0.0072. 


We  may  also  compute  the  weights  by  means  of  the  equations  (96). 
Thus,  to  find  the  weight  of  y,  we  have 


and  hence 


ldd.2l  =  [c?(/.l]  —  ^^^  Icd.l]  =  +  0.02977, 

^.=E^ -eft*"! =»■»»'*■ 


The  equations  (103)  and  (108)  are  convenient  for  the  determination 
of  the  values  and  weights  of  the  unknown   quantities   separately. 


CORRECTION  OF  THE   ELEMENTS. 


421 


Thus,  by  means  of  the  values  of  tlie  auxiliaries  obtained  in  the  iirst 
elimination,  we  find  from  the  equations  (100),  (101),  and  (102), 

A"  =  +  16.r)442, 
B"'  =  +    0.1202, 


A'  =  —  2.9052, 
B"  =--  —  4.5G91, 


A'" 
C" 


—  3.3012, 

—  0.9356, 


and  then  the  equations  (103)  and  (108)  give 


x--=  —  81".609,       y : 
p^  ^  0.00057,         p^ 


+  23".977,       2  =  —  2".705,      «  =  +  17".316, 
0.0074,  p^  =.  0.0312,         ^„  =:.  0.0297, 

agreeing  with  the  results  obtained  by  means  of  the  other  methods. 
The  weights  arc  so  small  that  it  may  be  inferred  at  onee  that  the 
values  of  x,  y,  z,  and  u  are  very  uncertain,  although  they  are  those 
whi(;h  best  satisfy  the  given  equations.  It  will  be  observed  that  if 
we  multiply  tlic  first  normal  equation  by  2.9,  the  resulting  equation 
will  diifer  very  little  from  the  second  normal  equation,  and  hence  we 
have  nearly  the  case  presented  in  which  the  number  of  independent 
relations  is  one  less  than  the  number  of  unknown  quantities. 

The  uncertainty  of  the  solution  will  be  further  indicated  by  deter- 
mining the  probable  errors  of  the  results,  although  on  account  of  the 
small  number  of  equations  the  probable  or  mean  errors  obtained  may 
bo  little  more  than  rude  approximations.  Thus,  adopting  the  value 
of  [vv]  obtained  by  direct  substitution,  we  have 


and  hence 


./l»'±i]  =  jpl|  =  2.416, 


1".629, 


which  is  the  probable  error  of  the  absolute  term  of  an  equation  of 
cond'^^'on  whose  weight  is  unity.     Then  the  equations 


r 


r..  = 


r  r        g 

— =,  r^  =  ~^,  &c., 

Vp^  Vp, 


give 
r^=±68".25. 


r  =:  ±  18".94,        r^  =  d=  9".22, 


±  9".45. 


It  thus  appears  that  the  probable  error  of  s  exceeds  the  value  obtained 
for  the  quantity  itself,  and  that  although  the  sum  of  the  squares  of 
the  residuals  is  reduced  from  204.31  to  11.67,  the  results  are  still 
quite  uncertain. 

153.  The  certainty  of  the  solution  will  be  greatest  when  the  coef- 
ficients in  the  equations  of  condition  and  also  in  the  normal  equations 


422 


TIIEOUKTICAI.    ASTHOXOMY. 


/ 


dilfor  very  considoriibly  both  in  niaj^iiitudo  and  in  sign.  In  tlie  cor- 
rection of  tlio  elements  of  the;  orbit  of  a  planet  when  the  observa- 
tions extend  only  over  a  short  interval  of  time,  the  coeflieienls  will 
generally  change  value  so  slowly  that  the  ecjnations  for  the  direct 
determination  of  the  corrections  to  bo  aj)})lied  to  the  elements  will 
not  atford  a  satisfactory  solution.  In  such  cases  it  will  be  expedient 
to  form  the  eciuations  for  the  determination  of  a  less  number  of 
(juantities  from  which  the  corrected  elements  may  be  subsequently 
derived.  Thus  we  may  determine  the  corrections  to  be  applied  to 
two  assumed  geocentric  distances  or  to  any  other  quantities  which 
afford  the  required  convenience  in  the  solution  of  the  problem, 
various  formula;  for  which  have  been  given  in  the  preceding  chapter. 
The  (quantities  selected  for  correction  should  be  known  functions  of 
the  elements,  and  such  that  the  equations  to  be  solved,  in  order  to 
combine  all  the  observed  places,  shall  not  be  subject  to  any  uncer- 
tainty in  the  solution.  But  when  the  observations  extend  over  a  long 
period,  the  most  complete  determination  of  the  corrections  to  be 
applied  to  the  provisional  elements  will  be  obtained  by  forming  the 
equations  for  these  variations  directly,  and  combining  them  as  already 
explained.  A  complete  proof  of  the  accuracy  of  the  entire  calcula- 
tion will  be  obtained  by  computing  the  normal  places  directly  from 
the  elements  as  finally  corrected,  and  comparing  the  residuals  thus 
derived  with  those  given  by  the  substitution  of  the  adopted  values 
of  the  unknown  quantities  in  the  original  equations  of  condition. 

If  the  elements  to  be  corrected  differ  so  much  from  the  true  value? 
that  the  squares  and  jn'oducts  of  the  corrections  are  of  sensible  maj^- 
nitude,  so  that  the  assumption  of  a  linear  form  for  the  equations  does 
not  afford  the  required  accuracy,  it  will  be  necessary  to  solve  the 
etpiations  first  provisionally,  and,  having  applied  the  resulting  cor- 
rections to  the  elements,  we  compute  the  places  of  the  body  directly 
from  the  corrected  elements,  iind  the  differences  between  these  and 
the  observed  places  furnish  '.lew  values  of  n,  n',  n",  etc.,  to  be  used 
in  a  repetition  of  the  solution.  The  corrections  which  i-esult  from 
the  second  solution  will  be  small,  and,  being  applied  to  the  eleineiit.s 
as  corrected  by  the  first  solution,  will  furnish  satisfactory  results.  In 
this  new  solution  it  will  not  in  general  be  necessary  to  recompute  the 
coefficients  of  the  unknown  quantities  in  the  equations  of  condition, 
since  the  variations  of  the  elements  will  not  be  large  enough  to  affect 
sensibly  the  values  of  their  differential  coefficients  with  respect  to 
the  observed  spherical  co-ordinates.  Cases  may  occur,  however,  in 
which  it  may  become  necessary  to  recompute  the  coefficients  of  one 


CORRFXTIOX   OF   THE   J^jEMEXTS. 


423 


or  more  of  tlie  unknown  quantities,  hut  only  when  tla'so  coolVicicnt?! 
arc  vorv  consiilcriibly  cluingcd  by  ii  small  variation  in  tlio  udopti'd 
values  of  the  oloments  employed  in  the  calculation.  In  such  cases 
the  residuals  obtained  by  sui)stitution  in  the  equations  of  condition 
will  not  agree  with  those  obtained  by  direct  calcidation  unless  the 
C(»rrections  applied  to  the  corres])onding  elements  are  very  small.  It 
may  also  be  remarked  that  often,  and  especially  in  a  repetition  of  the 
solution  so  as  to  include  terms  of  the  second  order,  it  will  l)e  sufli- 
ciently  accurate  to  relax  a  little  the  rigorous  requirements  of  a  com- 
plete solution,  and  use,  instead  of  the  actual  coetHcients,  ecpiivalent 
numbers  which  are  more  convenient  in  the  numerical  operations  re- 
quired. Although  the  greatest  confidence  should  bo  })laced  in  the 
accuracy  of  the  results  obtained  as  far  as  possible  in  strict  accordance 
with  the  requirements  of  the  theory,  yet  the  uncertainty  of  the  deter- 
mination of  the  relative  weights  in  the  cond)ination  of  a  series  of 
observations,  as  well  as  the  effect  of  uncliminated  constant  errors, 
may  at  least  warrant  a  little  latitude  in  the  numeric^il  application, 
provided  that  the  weights  of  the  I'esults  are  not  thereby  much  affected. 
A  constant  error  may  in  fact  be  regarded  as  an  unknown  <[Uantity  to 
he  determined,  and  since  the  effect  of  the  omission  of  one  of  the 
unknown  (pmntities  is  to  diminish  the  probable  errors  of  the  resulting 
values  of  the  others,  it  is  evident  that,  on  account  of  the  existence  of 
constant  errors  not  determined,  the  values  of  the  variables  obtained 
l)y  the  method  of  least  squares  from  different  corresponding  series  of 
ol)servations  may  differ  beyond  the  limits  which  the  probable  errors 
of  the  different  determinations  have  assigned.  Further,  it  should  be 
ol)served  that,  on  account  of  the  unavoidable  uncertainty  in  the  esti- 
mation of  the  weights  of  the  observations  in  the  preliminary  cond)i- 
nation,  the  probable  error  of  an  observed  place  whose  weight  is 
unity  as  determined  by  the  final  residuals  given  by  the  equations  of 
con<lition,  may  not  agree  exactly  with  that  indicated  by  the  prior 
discussion  of  the  observations. 


154.  In  the  case  of  verv  eccentric  orbits  in  which  the  corrections 
to  be  applied  to  certain  elements  are  not  indicated  with  certainty  by 
the  observations,  it  will  often  become  necessary  to  make  that  whose 
weight  is  very  small  the  last  in  the  elimination,  and  determine  the 
other  corrections  as  functions  of  this  one:  and  whenever  the  coeffi- 
cients of  two  of  the  unknown  quantities  are  nearly  ecpial  or  have 
nearly  the  same  ratio  to  each  other  in  all  the  different  e(piations  of 
condition,  this  method  is  indispensable  unless  the  difficulty  is  reme- 


424 


TIIEOIIETICAL   ASTRONOMY. 


(lied  by  other  means,  sucli  as  the  iiitro(hictioii  of  different  elements  or 
diflerent  combinations  of  the  same  elements.  The  eqiiatioiiH  (113) 
fiu'nish  the  values  of  the  unknown  quantities  when  wo  ne^^leet  that 
which  is  to  bo  determined  independently;  and  then  the  e([uati()ii.s 
(114)  give  the  required  expressions  for  the  complete  values  of  those 
quantities.  Thus,  when  a  comet  has  been  observed  only  during  u 
brief  period,  the  elliptieity  of  the  orbit,  however,  being  plainly  indi- 
cated by  the  observations,  the  determination  of  the  correction  to  be 
applied  to  the  mean  daily  motion  as  given  by  the  provisional  ele- 
ments, in  connection  with  the  corrections  of  the  other  elements,  will 
necessarily  be  quite  uncertain,  and  this  uncertainty  may  very  gmuly 
affect  all  the  results.  Hence  the  elimination  will  be  so  arranged  that 
A/i  shall  be  the  last,  and  the  other  corrections  will  be  determined  as 
functions  of  this  quantity.  The  substitution  of  the  results  thus 
derived  in  the  equations  of  condition  will  give  for  each  residue'.  »n 
expression  of  the  following  form : — 


Thei'efore  we  shall  have 


Atf  =  V„  -j-  yAn. 


\.vV]  =  [^vf'o]  +  2  lv,rl  ^//  +  [rrl  am',  (137) 

which  may  be  applied  more  conveniently  in  the  equivalent  form 

W  =^  [Vo]  -  ^[^^  CV]  +  M  (  AM  +  ^  )\  (138) 


The  most  probable  value  of  A/i  will  be  that  >\diich  renders  [vv]  a 
minimum,  or 

M' 


Afi^ 


(139) 


and  the  corresponding   value  of  the   sum  of  the  squares  of  the 
residuals  is 


M  =  Iv^vJ  -  ^  M- 


(1-10) 


The  correction  given  by  equation  (139)  having  been  applied  to  u, 
the  result  may  be  regarded  as  the  most  probable  value  of  that  cle- 
ment, and  the  corresponding  values  of  the  corrections  of  the  other 
elements  as  determined  by  the  equations  (114)  having  been  also  duly 
applied,  we  obtain  the  most  probable  system  of  elements.  These, 
however,  may  still  be  expressed  in  the  form 


Si   +  ^O^/i, 


i  +  Bo^n, 


r.  -f  CflAAi,  &C. 


CORRECTION   OF  THE   ELEMENTS. 


425 


the  coofficiciits  Ag,  ^„,  C'u,  S:r.  bcin<;  tlioso  givoii  by  the  o(iuations 
(114),  and  thus  the  eh'inents  may  he  derived  which  correspond  to  any 
assumed  value  of  //  dillerinj^  iVom  its  most  probable  vahie.  The 
unknown  (|uantity  A/i  will  also  be  retained  in  the  values  of  the 
residuals.  Hence,  if  we  assign  small  increments  to  /i,  it  may  easily 
be  seen  how  much  this  element  may  ditler  from  its  most  probable 
value  without  giving  results  for  the  residuals  which  are  incompatible 
with  the  evidence  furnished  by  the  observations. 

If  the  dimensions  of  the  orbit  are  expressed  by  means  of  the  ele- 
ments (J  and  e,  it  may  occur  that  the  latter  will  not  be  determined 
with  certainty  by  the  observations,  and  hence  it  should  be  tnvited  as 
suggested  in  the  case  of  /a;  and  we  proceeti  in  a  similar  manner  when 
the  correction  to  be  applied  to  a  given  value  of  the  semi-transverse 
axis  a  is  one  of  the  unknown  quantities  to  be  determined. 


420 


tiiix)Ui;tical  astkoxomy. 


CHAPTER  Vlir. 


/ 


INVESTIGATION  OK   VARIOUS   FORMt.'L.i:  KOU  THK   DKTKRMINATION  OK  THE  SPECIAL 
PERTURBATIONS   OF   A    IIEAVENIA'   iiODV. 

155.  We  liiivc  thus  fur  considered  the  oiroumstnnees  of  tlie  uiidis- 
tiirhed  motion  of  the  lieavenly  bodies  in  their  orhits;  l)ut  si  eoini)I('te 
determination  of  the  elements  of  the  orbit  of  any  body  revolving 
around  the  sun,  requires  that  wo  should  determine  the  alterations  in 
its  motion  due  to  the  aetion  of  the  other  bodies  of  the  system.  For 
this  purpose,  we  shall  resume  the  general  equations  (18),,  namely, 


dt 


'^  +  /c\l  +  m)~--=^kKl-\-ni-) 


''^U^(i  +  m)^  =  m+^^'^, 


(It 

d'z 


u) 


(Hi 


wliich  determine  the  motion  of  a  heavenly  body  relative  to  tiiO  sun 
when  subjeet  to  the  action  of  the  other  bodies  of  the  system.  We 
have,  further, 

m'     11       ■.cx'+yjf  +  zz'\  m"     (I       xx"  +  yf  +  zz" \  , 

which  is  called  iha  perturbinrj  fimdion,  of  which  the  partial  difforou- 
tial  coefficients,  with  respect  to  the  co-ordinates,  are 


dP.  _    m'    Ix'  —  x  /\          m!'    lx"—x  x"  \ 

d^  ~~  1  +  m \ '  p'  r"  /  +  1  +  «i \  '"p"  /''  I  "^        ' 

dy~l^  m  \     p'  r"  /  "^  1  +  )H  \     p"  r'"  j  ^        ' 

dQ  _    m'    Iz'  —  z  3^\          vi"    lz"  —  z  £'  \ 

dz~l  +  m\     p'  r''  j'^  l  +  mV  p"  7"  }  +        ' 


(2) 


and  in  wiiich  m',  m",  &c.  denote  the  ratios  of  the  masses  of  the 
several  disturbing  planets  to  the  mass  of  the  sun,  and  m  the  ratio  of 
the  mass  of  the  disturbed  planet  to  that  of  the  sun.  These  partial 
differential  coefficients,  when   multiplied  by  F(l  +  ?/i),  express  the 


I'KirrrmtA'i'ioNs. 


427 


TIIK  SriXIAL 

^  the  midis- 
:  a  complete 
Y  rcvolviiiif 
Iterations  in 
■.stem.  For 
namely, 


uj 


!  to  ti.o  sun 
stem.     ^\  0 

l  +  ^^'o., 

al  (lift'oron- 

-  &c., 

-  Ac,      (2) 

-  &e., 

sses  of  tlie 
:he  ratio  of 
ie.se  partial 
express  the 


sutii  of  the  compoiicnt.s  of  the  distiirliiiii;'  force  resolved  in  directions 
]i;iral!cl  to  the  three  re(;taiij>iil:ir  axes  respectively. 

When  we  nei^le(!t  the  consideration  <tf  the  pertiirhutions,  the  general 
cipiations  of  motion  become 


(It 


(W 


:  +h\\-Vm) 


X 


+  p(i  +  "0-!" 


0, 

:(), 


(3) 


-^  +  ^Hl  +  m)A^O, 


(It* 


tlie  complete  integration  of  which  furnishes  as  arbitrary  constjints  of 
integration  the  six  elements  wiiich  determine  the  orbitnal  motion  (»f  a 
heavenly  body.  IJut  if  we  regard  these  elements  as  representing  the 
actual  orbit  of  the  body  for  a  given  instant  of  time  /,  and  conceive 
of  the  effect  of  the  disturbing  forces  due  to  the  action  of  the  other 
bodies  of  the  system,  it  is  evident  that,  on  account  of  ilie  change 
arising  from  the  force  thus  introduced,  the  body  at  anotiier  instant 
diilerent  from  the  first  will  be  moving  in  an  orbit  for  which  the 
cloMicnts  are  in  some  degree  ditlcrent  from  those  which  satisfy  the 
original  equations.  Altliough  the  action  of  the  disturbing  force  is 
continuous,  we  may  yet  regard  the  elements  as  unchanged  during  the 
clement  of  time  dt,  and  as  varying  only  after  each  interval  dt.  Let 
us  now  designate  by  /„  the  epoch  to  which  the  elements  of  the  orbit 
belong,  and  let  these  elements  be  designated  by  J/„,  r^,  J^„,  /„,  <;„,  and 
0,1 ;  then  will  the  equations  (."])  be  exactly  satisfied  by  niciuis  of  the 
exi)rcssions  for  the  co-ordinates  in  terms  of  these  rigoi'ously-constant 
elements.  These  elements  will  express  the  motion  of  the  body  sub- 
ject to  the  action  of  the  disturbing  forces  only  during  the  infinitesimal 
interval  dt,  and  at  the  time  t^^  +  dt  it  will  commence  to  describe  a 
new  orbit  of  which  the  elements  Avill  differ  from  these  constt\nt  ele- 
ments by  increments  which  are  called  the  pcrtnrbdf torn. 

According  to  the  principle  of  the  variation  of  parameters,  or  of 
the  constants  of  integration,  the  differential  equations  (1)  will  be 
satisfied  by  integrals  of  the  same  form  as  those  obtained  when  the 
.second  members  are  put  equal  to  zero,  provided  only  that  the  arbitrary 
constants  of  the  latter  integration  are  no  longer  regarded  as  pure 
constants  but  as  subject  to  variation.  Consequently,  if  we  denote  the 
variable  elements  by  M,  ~,  SI,  i,  c,  and  a,  they  will  be  connected 
with  the  constant  elements,  or  those  which  determine  the  orbit  at  the 
instant  t„,  by  the  equations  ., 


428 


THEORETICAL   ASTRONOMY. 


111   WllU'll 


ich 


clt 
U 
dt 

(IJ:  dt ' 


d.T 
dt 

dt 


dt,  a::::^a„       fj        ^^ 

&c.  denote  the  differential  coefficients  of  the  el 


dt   ''' 
^"  dt, 


(4) 


I 


mcnts  depending  on  the  disturbing  forces.  When  these  differential 
coeHieients  are  known,  we  niivy  determine,  by  simple  quadrature,  the 
perturl)ati()ns  oM,  drr,  &c.  to  be  added  to  the  constant  elements  in 
order  to  obtain  those  corresponding  to  any  instant  for  wliieh  the 
place  of  the  body  is  required.  These  differential  coefficients,  ]iow<>V('r, 
are  functions  of  the  partial  differential  coefficients  of  iJ  with  rospoi't 
to  the  elements,  and  before  the  integration  can  be  perforn'cd  it 
becomes  necessary  to  find  the  expressions  for  these  partial  difft  ential 
coefficients.  For  this  purpose  we  expand  the  function  iJ  into  a  con- 
verging -  ries  and  then  differentiate  each  term  of  this  series  relatively 
to  the  elements.  This  function  is  usually  developed  into  a  converg- 
ing series  arranged  in  reference  to  the  ascending  pov/ers  of  the  eccen- 
tricities and  inclinations,  and  so  as  to  include  au  indefinite  number 
of  revolutions;  and  the  final  integration  will  then  give  what  are 
called  the  absolute  or  general  perturbatlo)is.  When  the  eccentricities 
and  irdinations  are  very  great,  as  in  the  case  of  the  comets,  thir? 
development  and  analytical  integration,  or  quadniture,  becomes  no 
longer  possible,  and  even  when  it  is  possible  it  may,  on  account  of 
the  magnitude  of  the  eccentricity  or  inclination,  become  so  dittifnilt 
that  we  are  obliged  to  determine,  instead  of  the  a1)solute  perturbations, 
what  are  called  the  Hpccial  perturbatiom,  by  nittiiods  oC  approyiiu.i- 
tion  known  as  mechanical  quadratures,  acfording  to  which  wo  deter- 
mine the  variations  of  the  elements  from  one  epoch  ("^  to  another 
epoch  f.  This  method  is  applicable  to  any  case,  and  may  be  advan- 
tageously employed  even  when  the  determination  of  the  absolute 
perturb;.! ions  is  possible,  and  especially  when  a  .series  of  observatioiirs 
extending  through  a  period  of  many  years  is  available  and  it  is 
desired  to  determine,  for  any  instant  t^^,  a  system  of  elements,  usuiillj' 
Called  osculating  element^',  ou  which  the  complete  theory  of  the  motion 
may  be  based. 

Instead  of  computing  the  variations  of  the  elements  of  the  orbit 
directly,  we  may  find  the  perturbations  of  any  known  functions  of 
these  elements;  and  the  most  direct  and  simple  method  is  to  deter- 
mine the  variations,  due  to  the  action  of  tlie  disturbing  forces,  of 
any  system  of  three  co-ordinates  by  means  of  which  the  position  of 


PERTURKATIOXS. 


429 


i 


ii;r!  body  or  the  elements  themselves  may  be  found.  We  shall,  there- 
fore, derive  various  Ibrmulie  for  this  purpose  before  investigating  the 
fbrniula)  for  the  direet  variation  of  the  elements. 

15'.).  Let  ;r„,  iJq,  z^^  be  the  reetangnlar  co-ordinates  of  the  body  at 
tlie  time  t  computed  b^  means  of  tiie  osculating  elements  J/„,  "„,  ^,|, 
&c.,  corresponding  to  the  cj)och  ^y.  Let  x,  y,  z  be  the  actual  co-ordi- 
nutcs  of  the  disturbed  body  at  the  time  t;  and  we  shall  have 

o.r,  '///,  and  dz  being  the  perturbations  of  the  rectangular  co-ordinates 
from  the  epot^h  ^^  to  the  time  t.  If  we  substitutf>  tliese  values  of  .c, 
)/,  and  z  in  the  erpuitions  (1),  aiid  then  subtract  from  each  the  corre- 
ipoiiding  one  of  equations  (3),  we  get 


dx' 


~df 


^+,.(x+.„)(!4i£-|)=*.a-,.)f. 


Let  us  now  put  r     -  r^  +  or;  then  to  terms  of  the  order  o)^,  which  is 
<i(juivalent  to  considering  oidy  the  first  powev  of  the  disturbing  force, 

ve  luivo 

'  'n  '0    ^  '0        ' 


and  hence 


dt'  d;/  r^        \    r^  '' f 


(6) 


d'Sz 

di:'' 


.a  +  »)-.«l^^(3^.-4 


Wo  have  also  from 

neglecting  terras  of  the  second  order, 


dr 


Vo 


2 ,5.r  4.  •.!5  hi  +  :i  ,h. 


(T) 


430 


THEORETICAL   ASTRONOMY. 


The  integration  of  the  eqiiutions  (6)  will  give  the  perturbations  nx, 
01/,  and  (h  to  be  applied  to  the  rectangular  co-ordini;tes  x^,  i/^,  z^  com- 
puted l)y  means  of  the  osculating  elements,  in  order  to  find  the  acttiai 
co-ordinates  of  the  body  for  the  date  to  which  the  integration  belongs, 
But  since  the  second  members  contain  the  quantities  dx,  dy,  dz  wliich 
are  sought,  the  integration  must  be  effected  indirectly  by  successive 
approximations;  and  from  the  manner  in  which  these  are  invt)lve(l 
in  the  second  members  of  the  equations,  it  will  ap])ear  that  this  inte- 
gration is  possible. 

If  we  consider  only  a  single  disturbing  planet,  according  to  the 
equations  (2),  wo  shall  have 


P(l+»u) 


dz 


(3) 


and  these  forces  wc  will  designate  by  A'^  Y,  and  Z  respectively ;  then, 
if  in  these  expressions  we  neglect  the  terms  of  the  order  of  the 
square  of  the  disturbing  force,  writing  .r^,  «/„,  s,,  in  place  of  x,  i/,  :, 
the  equations  (6)  become 


d^'i'ix 


=  X„  + 


k'(l-\-vi) 


'o  \     '0  ' 


df 
d}ori 
"(W 

dt'~    ''^  3 


(9) 


which  are  the  equations  for  computing  the  perturbations  of  the  rec- 
tangular co-ordinates  with  reference  only  to  the  first  power  of  the 
masses  or  disturbing  forces.     We  have,  furthei", 

/>'  =  (a/  -  xr  +(ij-  yY  +  (/  -  zy,  (10) 

in  which,  when  terms  of  the  second  order  are  neglected,  we  use  tlie 
values  .Tfl,  j/q,  2„  for  x,  y,  and  z  respectively. 

157.  From  the  values  of  5.r,  di/,  and  dz  computed  with  regard  to 
the  first  power  of  the  masses  we  may,  by  a  repetition  of  part  of  the 
calculation,  take  into  account  the  squares  and  products  and  even  tlie 
higher  powers  of  the  disturbing  forces.  The  equations  (5)  may  bo 
written  thus: — 


VARIATIOX   OF   CO-ORDIXATES. 


431 


(11) 


~di' 


z  + 


^'(1  + 


in  which  nothing  is  neglected.  In  the  application  of  these  formula?, 
as  soon  as  ox,  dy,  and  oz  have  b?en  found  for  a  few  successive  inter- 
vals, we  may  readily  derive  approximate  values  of  these  quantities 
for  the  date  next  followiiig,  and  with  these  find 

and  hence  the  complete  values  of  the  forces  X,  Y,  and  Z,  by  means 
of  the  equations  (8).     To  find  an  expression  for  the  factor 

1  _^ 

which  will  be  convenient  in  the  numerical  calculation,  we  have 

r«  =  (x,  +  dxy  +  (y,  +  dyy  +  (z,  +  .hy 

=  r,^  +  2x,.^x  +  2ij,<^y  +  2z,oz  +  d.c^  +  ,hf  +  dz\ 

and  therefore 


1  =  1  +  2 


Cro+^^^O'^-^- +  (?/,+  A'5.y)  %  +  (gp  +  ■  >^  -J? 


Let  us  now  put 

'o  'o  'o 

and 


(12) 


/9  =  1 


=  l-(l  +  2<z)-^; 


then  we  shall  have 

f     Q / 1       5     ,5.7,      5 . 7 . 9  ,  ^  „     \ 


(13) 


and  the  values  of/  may  be  tabulated  with  the  argument  q.     The 
equations  (11)  therefore  become 


d'l^x 


dt 


r  =  X-^ 


U'{\-\-m) 


».» 


{fqx  —  f5.c), 


(14) 


Ill 


432 


THEORETICAL   ASTRONOMY. 


The  eoc'fticicnis  of  dx,  (Ji/,  and  oz  in  equation  (12)  may  be  found  at 
once,  with  sufliciont  accuracy,  by  means  of  tlio  approximate  values 
oftho.se  quantities;  and  having  found  the  value  of /corresponding 

to  the  rcsultin<^' value  01  q,  the  numerical  values  oi     ,„-.     ;.r>  and 

-.,.,-,  which  include  the  squares  and  products  of  the  masses,  will  he 

obtained.     The;  integration  of  these  will  give  more  exact  values  of 

S.V,  di/,  and  oz,  and  then,  recomputing  q  and  the  other  quantities  which 

require  correction,  a  still  closer  approximation  to  the  exact  values  of 

the  })erturl)ations  will  result. 

Tabh  XVII.  gives  the  values  of  log/  for  positive  or  negative 

values  of  q  at  intervals  of  0.000 1  from  (/=  (i  to  ry -~- 0.03.     Unless 

the  perturl)ations  are  very  large,  q  will  be  found  within  the  limits  of 

this  table;  and  in  those  cases  in  which  it  exceeds  the  limits  of  the 

table,  the  value  of 

r'" 

r 


Sq  =  ^ 


'  0 


may  be  computed  directly,  using  the  value  of  r  in  terras  of  r^  and 
dx,  OI/,  dz. 

In  the  application  of  the  preceding  forraulffi,  the  positions  of  tiie 
disturbed  and  disturbing  bodies  may  be  referred  to  au}'  system  of 
rectangular  co-ordinates.  It  will  be  advisable,  however,  to  adopt 
either  the  plane  of  the  equator  or  that  of  the  ecliptic  as  the  futida- 
mental  plane,  the  positive  axis  of  x  being  directed  to  the  vernal 
equinox.  By  choosing  the  plane  of  the  elliptic  orbit  at  the  time  /,, 
as  the  plane  of  xy,  the  co-ordinate  ;:  will  be  of  the  order  of  the  per- 
turbations, and  the  calculation  of  this  part  of  the  action  of  the  dis- 
turbing force  will  be  very  much  abbreviated;  but  unless  the  inclina- 
tion is  very  large  there  will  be  no  actual  advantage  in  this  selection, 
since  the  computation  of  the  values  di'  the  components  of  the  dis- 
turbing forces  will  require  more  labor  than  when  either  the  e([uator 
or  the  ecliptic  is  taken  as  the  fundamental  plane.  The  perturbations 
computed  for  one  fundamental  plane  may  be  converted  into  those 
referred  to  another  plane  or  to  a  different  position  of  the  axes  in  the 
same  plane  by  means  of  the  formulae  which  give  the  transformation 
of  the  co-ordinates  directly. 

158.  We  shall  now  investigate  the  formula)  for  he  integration  of 
the  linear  diifercntial  equations  of  the  second  order  which  express  tlie 
variation  of  the  co-ordinates,  and  generally  the  furmulaj  for  finding 

the  integrals  of  expressions  of  the  form  j  /(.r)  dx  and  j  I  f{x)  dx^ 


MECHANICAL  QUADRATURE. 


433 


when  the  values  of /(.c)  are  computed  for  successive  values  of  ;r  in- 
creasing in  arithmetical  progression.     First,  therefore,  wo  shall  find 
the  integral  of /(.r)  d.v  ',vithin  given  limits. 
Witiiin  the  limits  for  which  x  is  continuous,  we  have 


fix)  =  a  +  /3.C  +  yx''  +  (Jar"  +  ex*  + 


(15) 


and  if  we  consider  only  three  terras  of  this  series,  the  resulting  equa- 
tion 

/(a;)  —  a  +  /J.r  +  yx" 

is  that  of  the  common  parabola  of  which  the  abscissa  is  x  and  the 
ordinate /(.r),  and  the  integral  of /(a;)  dx  is  the  area  included  by  the 
abscissa,  two  ordinates,  and  the  included  arc  of  this  curve.  Gene- 
rally, therefore,  we  may  consider  the  more  complete  expression  for 
/(.(•)  as  the  equation  of  a  parabo'ic  curve  whose  degree  is  one  less 
tlian  the  number  of  terras  taken.  Hence,  if  we  take  n  terras  of  the 
scries  as  the  value  of/(;c),  we  shall  derive  the  equation  for  a  parabola 
whose  degree  k  n~  1,  and  which  has  n  points  in  common  with  the 
curve  represented  by  the  exact  value  oi'  f{x). 

If  we  multiply  equation  (15)  by  dx  and  integrate  between  the 
limits  0  and  x',  wc  get 


J  fix)  dx  ^  ax'  -f  ^iix'^  +  Ir^'  4-  -Idx'*  +  . . 


(16) 


If  now  the  values  of  f{x)  for  different  values  of  x  from  0  to  x'  are 
kiuiwn,  each  of  these,  by  means  of  equation  (15),  will  furnish  an 
('([Uiition  for  the  determination  of  a,  /?,  y,  &c. ;  and  the  number  of 
terms  which  may  bo  taken  will  be  equal  to  the  number  of  different 
known  values  oi'  f{x).  As  soon  as  a,  /?,  y,  &c.  have  th  'i  been  found, 
the  equal luii  (16)  will  give  the  integral  required. 

If  tlie  values  of /(.f)  are  computed  for  values  of  x  at  equal  inter- 
vals and  we  integrate  between  the  limits  x  =  0,  and  x  =  riAx,  Aa' 
l)oiiig  the  constant  interval  between  the  successive  values  of  x,  and 
11  the  number  of  intervals  from  the  beginning  of  the  integration,  we 
obtain 

f{x)  dx  ^  anux  +  ]/J/i'A.c»  4-  ?^rn^^3^  +  &c. 

0 

Let  us  now  suppose  a  quadratic  paral)ola  to  pass  through  the  points 
of  tlie  curve  represented  by  f{x),  corresponding  to  x  —  0,  x 

28 


/^ 


£kX, 


f 


434 


THEORETICAL  ASTROXOMY. 


and  X  =  2a;k/  then  will  the  area  included  by  the  arc  of  this  parabola, 
the  extreme  ordinates,  and  the  axis  of  abscissas  be 


2Aj; 


Cfix)  (Ix  =  Aa;  (2a  +  2,?Aa;  +  ir^x"). 

0 

The  equation  of  the  ;  urve  gives,  if  we  designate  the  ordinates  of  the 
three  successive  jwints  ;;y  y^^,  y^,  and  y^, 


2ax 


and  hence  wc  derive 


2Aa! 

ffix)dx 


2ax' 


^x  (2/0  +  4?/,  +  y,). 


In  a  similar  manner,  the  area  included  by  the  ordinates  3/2  and  y^, — 
corresponding  to  x  ==  2ax  and  x  =--  4^x, — the  axis  of  abscissas,  and 
the  parabola  passing  through  the  three  points  corresponding  to  t/jj  ^3) 
and  y^,  is  found  to  be 


4Ax 


fjXic)  dx  =  J  Aa;  (y,  +  4y^  +  y^) ; 

2Sx 


and  hence  we  have,  finally, 


nAx 


(10 


Jf(x)  dx  ==  J  Ax  (y„_,  +  42/,, _i  +  yj. 

(H  — 2)Ax 

The  sum  of  all  these  gives 

nSx 

ffix)dx 

0 

=  J^-^-((2/o  +  2/J +  4(2/, +  2/3+2/5  + .  "^/n-i) +  2(2/j  +  ^*  + .  ..y„-.)), 

by  means  of  which  the  approximate  value  of  the  integial  within  the 
given  limits  may  be  found. 

If  we  consider  the  curve  which  passes  through  four  points  corre- 
sponding to  2/u,  2/1, 2/2>  aiid  2/3)  we  have 

y  =f(,x)  =  tt  +  /?a;  +  yx*  +  ^ar" 

for  the  equation  of  the  curve,  and  hence,  giving  to  x  the  values  0, 
A,r,  2a^,  and  Sao;,  successively,  we  easily  find 


MECHANICAL   QUADRATURE. 


435 


nates  of  the 


6A.r 
Therefore  we  shall  have 

3Ax 


"'  =  ft  ^:;5  (2/3  —  3(/2  +  3)/,  —  2/0). 


J/(a;)  rfj;  =  I  AX  (y„  +  3^^  +  3y,  +  2/3). 


(18) 


In  like  manner,  by  taking  successively  an  additioniil  terra  of  the 
series,  Ave  may  derive 


4A« 

ffix)  dx  =  ^^^-  Cly,  +  32^,  +  12i/,  +  32^3  +  1yd, 

6Ak 

//(.t-)  dx  =  1^  (1%„  +  75y,  +  50y,  +  0O//3  +  Iby,  +  10^5). 


(19) 


This  process  may  be  continued  so  as  to  include  the  extreme  values  of 
X  for  which /(a;)  is  known;  but  in  the  calculation  of  perturbations  it 
will  be  more  convenient  to  use  the  finite  differences  of  the  function 
instead  of  the  function  itself  directly.  Wc  may  remark,  further, 
that  the  intervals  of  quadrature  when  the  function  itself  is  used, 
may  be  so  determined  that  the  degree  of  approximation  will  be  much 
greater  than  Avhen  these  intervals  are  uniform. 

159.  Let  us  put  A.r  =  to,  and  let  the  value  of  x  for  which  71  =  0 
be  designated  by  a;  then  will  the  geucrai  value  be 

(0  being  the  constant  interval  at  which  the  values  of  f{x)  are  givcu. 
Honce  we  shall  have 

dx  =  (odn, 
I  /(*)  dx  =  u)l  /(a  +  nut)  dn. 

If  we  expand  the  function /(a  +  nco),  we  have 


/(a  +  n.)=/(a)  +  n^-^^.  +  ^--^.  -1-^  +  ^-^  . -i -^  +  fee.  (20) 


THEORETICAL  ASTRONOMY. 


1  n  ■ 


■/ 


436 

and  hence 

J /(a  -f-  nw)  dn  =  C  ■{-  nf(a)  +  h 


da 


+  ^' 


l«'/(i^ 


d^fia) 
da' 


+  .V^V'^  +  &c.. 


(21) 


C  being  the  constant  of  integration.     The  equations  (54)g  give 


w 


da 

dy(a) 

da' 


-  =/'"  (a)  -  1/^ :«)  +  liu/'"  («)-..., 


„,dy(a) 

da^ 
da^ 


=r(«)-i/"'(«)+.-., 
=r(«)-ir  "(«)  +  •••> 


(22) 


in  whicJi  the  functional  syml)ols  in  the  second  members  denote  the 
difterent  orders  of  finite  differences  of  the  function.    Hence  we  obtain 

I  /(a  4"  nut)  dn  =  C  +  r</(a) 

+  l'^H/'(«)  -  oV'"(«)  +  i^n<^)  -  Tla/^"(«)  +  •  •  •) 
+  J"' (/"(«)  -  i^/'^(a)  +  ^a/''(«)  -  5io/^"'(«)  •  H  . . .) 

+  .Ln*(.r(«)-:[r(«)  +  T?or'^«)-.--)  ,oo^ 

+  f  kn»  (r  (a) -ir '(«)  +  ...) 
+  W4o'^^(r(«)-^-r'"(«)  +  .--) 

If  M'e  take  the  integral  between  the  limits  —n'  and  -\-n',  the  terms 
containing  the  even  powers  of  n  disappear.  Further,  since  the  values 
of  the  function  are  supposed  to  be  known  for  a  series  of  values  of  n 
at  intervals  of  a  unit,  it  will  evidently  be  convenient  to  determine 
the  integral  between  the  required  limits  by  means  of  the  sum  of  a 
series  of  integrals  whose  limits  are  successively  increased  by  a  unit, 
such  that  the  difference  between  the  superior  and  the  inferior  limit 
of  each  integral  shall  be  a  unit.  Hence  we  take  the  first  integral 
between  the  limits  —\  and  +^,  and  the  equation  (23)  gives,  after 
reduction, 


MECHANICAL  QUADRATURE. 


437 


+  * 


^-i  (24) 

It  is  evidont  that  by  writing,  in  succession,  a  +  to,  o.  +  2co, .... 
o  +  Uo  in  place  of  a,  we  simply  add  1  to  each  limit  successively,  so 
that  we  have 


<  +  t 


+  i 


I  f(a  +  nw)  rfu  =  I  /((a  +  i^)  +  ('^  —  0  <")d(n  —  i) 

i-i  -i 

But  since 


«•  +  * 


i 


J 


i  +  i 


i  /(«  +  no})  dn  =  I  /(a  4"  »i<")  of>*^  +  I  /(«  +  'i*")  (?H -\-i  f(a-^niw)  dn, 

-k  -4  1  i-i 

if  we  give  to  i  successively  the  values  0,  1,  2,  3,  &c.  in  the  preceding 
equation,  and  add  the  results,  we  get 

i  +  i  n  —  i  n  =  i 

J/(a  +  nco)  dn  =  ^/(a  -f  nm)  +  ^\  ^/"  («  +  n^) 

n  =  0  7!  =  0 

Let  us  now  consider  the  functions  /(«),  /(«  +  nco),  &c.  as  being 
thoniselvcs  the  finite  differences  of  other  functions  symbolized  by  '/, 
tlio  first  of  which  is  entirely  ai'bitrary,  so  that  we  may  put,  in  accord- 
ance with  the  adopted  notation, 

/(aJ='/(a  +  >)-'/(a->), 
/(a  +  ^)  =  '/(a  +  !">)  -  7(«  +  -», 

/(a  +  n<o)  =  '/(a  +  {n  +  ^)  «>)  -  '/(a  +  (n  -  i)  <o). 
Therefore  we  shall  have 

n  =  i 

^/(a  +  nm)  =  '/(a  +  ( j  +  -i)  a,)  -  '/(«  -  », 


and  also 


n  =  0 


2/"  («  +  no*)  =/'  (a  +  (i  +  -V  «')  -/  (a  -  A<-), 

n  =  i 

^ria  +  no>)  =/"'(a  +  (i  +  ^)«^)  -/'"  (a  -  >),  &c. 


n  =  0 


438 


TIIEOUETICAL   ASTRONOMY. 


Furtlior,  since  tlio  quantity  'f{<i.  —  \o))  is  entiroly  arbitrary,  we  may 
assign  to  it  a  valne  such  that  the  sum  of  all  the  terms  of  tlie  equation 
which  have  the  argument  a  —  \io  siiall  be  zero,  namely, 

y(a~4«>)=:-,'i/'(a-H  +  5^L/"'(«->)-«j^V^or  («-•'•)+<&€. 

(20) 
Substituting  these  values  in  (25),  it  reduces  to 


o  +  (*'  -\-  i)«> 


i  +  1 


I  f{x)  rfx  =  w  I  /(a  -{■  niu)  dn 

=  wi'/(a  +  (i  +  ^.)«>)  +  .'J'(«  +  (/+P'-)  ^^'^ 

-sao/"'(«+('+i)-)+WVi5^o/''(«+('-l-P"')-»^<'-! 

In  the  calculation  of  the  perturbations  of  a  heavenly  body,  the 
dates  for  which  the  values  of  the  functioji  arc  computed  may  be  so 
ai'ranged  that  for  7i  =  —  .|,  corresponding  to  the  inferior  limit,  the 
integral  shall  be  equal  to  zero,  the  epoch  of /(«  —  \io)  being  that  of 
the  osculating  elements.  It  will  be  observed  that  the  equation  (26) 
expresses  this  condition,  the  constant  of  integration  being  included 
in  '/(a  —  \io).  If,  instead  of  being  equal  to  zero,  the  integral  has  a 
given  value  when  n  =  —  \y  it  is  evidently  only  necessary  to  add  this 
value  to  '/(a  —  \io)  as  given  by  (26). 

160.  The  interval  lo  and  the  arguments  of  the  function  may  always 
be  so  taken  that  the  equation  (27)  will  furnish  the  required  integral, 
either  directly  or  by  interpolation ;  but  it  will  often  be  convenient  to 
integrate  for  other  limits  directly,  thus  avoiding  a  subsequent  inter- 
polation. The  derivation  of  the  required  formulae  of  integration 
may  be  effected  in  a  maimer  entirely  analogous  to  that  already  indi- 
cated. Thus,  let  it  be  required  to  find  the  expression  for  the  integral 
taken  between  the  limits  — \  and  i. 

The  general  formula  (23)  gives 

4 
J/(a  +  n«>)dM  =  i/(a)  +  ^/(a)  +  5<g/"(a)-3g5/'"(a)  -  ttV3o/''(«) 

and  since,  according  to  the  notation  adopted, 

/' (a)  -  U/' («-»+/(«  + ^'")) 
=/'(«  + W     -A/"  (a), 


&c. ; 


/'"(a)::=r(a-i-» 
r(a)=r(a  +  i"') 


-ir(a). 

-^r(«),«fec., 


(28) 


MECHANICAL   QUAUHATURE. 


431) 


this  becomes 


jA«+«-)rf«=.i/(«)+^/'(«+.i-)-,'.7''(«)-3i,r'(«+.i-)    (29) 

+  tHu/'"  («)  +  :fif;ior  («  +  »  -  73'.;Vflnr(«)  -  &C. 
Therefore  we  obtain 


»'  +  * 


(30.) 


Now  we  have 

/■(a  +  ?tw)  dn  =  I  /(a  -[-  nio)  dn  —  (  J\a  +  nto)  dn ; 


and  if  we  substitute  the  values  already  found  for  the  terms  in  the 
second  member,  and  also 


/(a  +  ia,)  =    'f{a  +  (/  +  .p  oj)  -    '/(a  +  (i  -  A)  .>), 

/'v(a  4-  ;•«>)  =/'"(«  +  (i  +  A)'")  -/'"(«  +  (i  -  i)<"), 
/>'(a  +  H  =  p{a  +  (i  +  Aj'")  -  r(a  +  (i  -  ^)w),  &c. 


(31) 


we  get 

a  +  I'ui  i 

I  fix)  dx  =  oji  J  {a  +  Hw)  dn 

a-iu  —  i  ("^2) 

-WoV5ur(«+(^+-D'")-WoVaur(«+(*-i)'")+&c.i, 

which  is  the  required  integral  between  the  limits  — |  and  i. 

161.  The  methods  of  integration  thus  far  considered  apply  to  the 

cases  in  whicii  but  a  single  integration  is  required,  and  when  appli'^d 

to  the  integration  of  the  differential  equations  for  the  variations  jf 

the  co-ordinates  on  account  of  the  action  of  disturbing  bodic",  thev 

.„,.,,         ,  „  ddx  d'hi        .  ddz  , 

will  only  give  the  values  oi  -5-,  -57.  and  -j-,  and  another  integration 

becomes  necessary  in  order  to  obtain  the  values  of  o.v,  dy,  and  3z. 
We  will  therefore  proceed  to  derive  formulse  for  the  determination 
of  the  double  integral  directly. 


440 


TUKOUKVIVAL   ASTRONOMY. 


For  the  double  integral  i  jf{x)(l.i;'  we  have,  since  dx-  ~  lohln^, 
fffix)  dx^  =--  uj'  fffia  +  nu>)  dn\ 

The  value  of  the  function  designated  by  /((/)  being  so  taken  that 
when  n  '-^  —  J, 

I  /(ffl  +  "'")  dn  =  0, 

the  equation  (23)  gives 

0 

C—  i  f(,a  +  noj)dn. 
Therefore,  the  general  e^^uation  is 


I  /("  +  "*")  dn  =  I  /(a  +  nut)  dn  -\-  nf(a) 

the  values  of  a,  ,9, ;',...  being  given  by  the  equations  (22).  Multi- 
plying this  by  dn,  and  integrating,  we  get 

0 

fffia  +  nio)  dn'  =  C"  +  nffXa  +  nu>)  dn  +  in'fia) 

C"  being  the  new  constant  of  integration.  If  we  take  the  integral 
between  the  limits  —  ^  and  -|-  I,  we  find 

Jfj/(a  +  na})dn'  =  Cf(a  +  nw)  dn  +  -^^a  +TM(jr  +  asssso^  +  &c. 
From  the  equation  (32)  we  get,  for  i  =  0, 

0 

ffia  +  nio)  dn  =  '/(a)  -  J./'  (a)  +  ^^Tif"  («)  -  5^1  hf  («)  +  &c.  (33) 
-i 

Substituting  this  value,  and  also  the  values  of  a,  y,  e,  &c., — which 
are  given  by  the  second  members  of  the  equations  (22), — in  the  pre- 
ceding equation,  and  reducing,  we  get 

+  i 
JJ/(a-|-nc«)rfn'=.'/(a)-,'J'(a)+^fJ,r'(a)-4?i|o/^(a)-f-&c.(34) 


MECHANICAL  QUADRATURE. 


441 


fff(a  +  "'")  '/'i'  ^  '/{a  +  tw)  —  :IJ'  (a  +  im) 


■  -1 
and 


n-i 


—  i  n  =■  0  n  ^-  0 


n-0 


(35) 


+  t1?u^/"(«  +  "<")  "  T!i^/y30^/''(«+  »'")  +  &C. 


n-O 


Wo  may  evitlently  consider  '/(a  —  Jw),  '/(a  +  Jro),  tfec.  as  the  ditfer- 
cnees  of  othex*  functions,  the  first  of  whicli  is  arbitrary,  so  that  we 
have 

'j(a)  =  A7(a  +  »  +  m«'  -  I'")  =  i7(a  +  ">)    -  hf"  («  -  '"). 

y(a  +  «;-^  =  iy(«  +  2«>)  +  Vfia  -f  »  =  Vy(a  +  2«>)  -  A/"  (a), 

-i7(«  +  0*-i)'4 

Thciofoic 

n  ---  0 

Hi 

n  -  i 
Tt  -  i 


n==0 


Substituting  these  values  in  equation  (35),  and  observing  that 

"/(«)  +  "fia  -  CO)  =  2"/(a  -  c.)  +    7(«  -  \<o\ 
/(a)  +   /(a  -  <")  =    2/(  a)  -  /  («  -  », 

/"  (a)  +/"  (a  —  w)  =  2/"  («)  -/"  («  —  .^./i),  &c., 

and  that,  since  "f{a  —  lo)  is  arbitrary,  we  may  put 

"/(a  -  CO)  =  ,V/(«)  -  ^^So  (2/"  («)  +  /"  («  -  «>))  ,„., 

+  1,8^55  (3/'K«)  +  2/"^ (a  -  0,))  -  Ac,         ^'^^^ 


442 


THEORETICAL    ASTRONOMY. 


tlie  integral  becoiaes 


a  -i-  (l  +  i)a) 


<-,  + 


a  —  i<u  —  i 


- '-■^  \  i "./■(«  +  (*  + 1) '")  +  V'/C"  +  ''^)  -  ..'h/(«  +  (^  + 1) "»)   (371 

-4'j(«+^"'"H-.iio.r(<*-KM-i)'-)4-,^i,,r(o-i-'''':^ 

Avhicli  is  the  expression  for  tlic  double  integral  between  tlie  lii.iits 
—  \  and  /  -j-  \. 

The  value  of  "f{a  —  (o)  given  by  equation  (30)  is  in  accordance 
with  the  sui)positio)i  that  tor  n  =^  —  ^  the  double  integral  is  equal  to 
zero,  and  this  condition  is  fuHilled  in  the  calculation  of  the  pcrti.r- 
bations  when  the  argument  a  —  |(o  corresponds  to  the  date  for  which 
the  osculating  elements  are  given.  If,  for  n~-  —  |,  neither  the  single 
nor  tile  double  integral  is  to  l)e  taken  equal  to  zero,  it  is  only  neces- 
sary to  add  the  given  value  of  the  single  integral  for  this  argunicnt 
to  the  value  of  'f(a  —  |w)  given  by  ecpiation  (26),  and  to  add  the 
given  value  of  the  double  integral  for  the  same  argument  to  the  value 
of  "/{a  —  co)  given  by  (30). 

102.  In  a  similar  manner  we  may  find  the  expressions  for  tlie 
double  integral  between  other  limits.  Thus,  let  it  be  required  to 
iind  the  double  integral  between  the  limits  — I  and  i. 

Between  the  limits  0  and  I  we  have 


4  'I 

jjf'"  -r  'i"')(^>i^  —  ■ijfid'  +  nu>)dn  +  |/(«)  +  4'go 


which  gives 


+  3i.V^5  + 


3  S  i  li 


r-\- 


4yobo 


'5  +  &c. 


4 


(38; 


5 1  JO 


/'^'(«)- 


3  r.  7 
3Ht01' 


-^r(' 


4-        '  S7 


gu/^'(")-i-'t(: 


and  this  again,  by  means  of  (28),  gives 


Sf 


i-ri 


fia  ~\-  «<«)  dn' ^ i'fia  +  (i  f  ^)  o,)  -^/(a  4- ;'■ )  - Vs/'  («  +  (^^4 ' ") 


+3^4.r(«+^«)-f5ijn,r(«-t-(^+y)'") 


37 


ia-rW^ 


i*i^^-/'("-l-('-l4)«')- 


I  ■'•< 


I '3  out) 


/V^J^af/<.)-|-^c. 


MECHANICAL   QUADKATURE. 


443 


cn  the  VuAiii 


Tlicreforc,  since 


i  +  i 


-i 


aiul 


'/(«  +  (i  +  A)w)  :■-  -  'y(a  +  (/  +  1  }o,)  -  "f(a  +  M, 
j" (rt  +  (i  +  A)  w)  z^    f{n  +  (  /  +  1 ) ...)  —    /(«  +  /r«), 

we  shall  have 

'',-;.  ^-^i  (-30) 

which  fiivos  the  ro(jnire(l  intcixral  between  the  limits  — \  and  /. 

lG."i.  It  will  be  observed  that  the  coetHoients  of  the  several  terms 
of  the  forninhe  of  intet^ration  eonverire  rapidly,  and  iK.'uee,  l)y  a 
jiroper  selection  of  the  interval  at  whieli  the  values  of  the  fiim^tion 
;up  eomputed,  it  will  not  bo  neees.sary  to  eon.sider  the  terms  whi(  h 
depend  on  the  fonrth  and  higher  orders  of  differences,  and  rarely 
tliusc  which  depend  on  the  second  and  third  differences.  The  value 
;i>si<ined  to  the  interval  to  must  be  such  that  we  may  interpolate  with 
certainty,  by  means  of  the  values  eomputed  directly,  all  values  of  the 
function  intermediate  to  the  extreme  limits  of  the  integration ;  and 
luiu'c,  if  the  fourth  and  higher  orders  of  difllerences  are  sensible,  it 
\vill  be  necessary  to  extend  the  direct  computation  of  the  values  of 
the  fii!iction  beyond  the  limits  which  would  otherwise  be  re([uir<Ml, 
ill  order  to  obtain  correct  values  of  the  differences  for  the  beginning 
and  end  of  ihc  integration.  It  will  be  ex])edient,  therefore,  to  take 
w. -(I  small  th.'it  the  fourth  and  higher  differences  may  be  neglected, 
Init  not  smaller  than  is  necessa'y  to  satisly  this  condition,  since  otlier- 
w>('  an  unn'icessary  amount  of  labor  would  be  expended  in  the 
direct  computation  of  the  values  of  the  function.  It  is  better,  how- 
ever, to  have  the  interval  w  smaller  than  what  would  appear  to  be 
strictly  re(piircd,  in  order  that  there  may  be  no  unceitainty  with 
respect  to  the  accuracy  of  the  integration.  On  account  of  the  rapidity 
with  which  the  higher  orders  of  diflerences  decrease  as  we  diminish 
w,  ;i  limit  for  the  magnitude  of  the  ado{)ted  intcj'val  will  s])eedily  be 
ol)t;iiiied.  The  magnitude  of  the  interval  will  therefore  be  suggested 
hy  the  rapidity  of  the  change  of  value  of  the  function.    In  the  com" 


I. 


444 


THEORETICAL   ASTUON<»MY. 


putation  of  the  perturbations  of  the  group  of  small  planets  between 
Mars  and  Jupiter  we  may  adopt  uniformly  an  interval  of  forty  days; 
but  in  the  determination  of  the  perturbations  of  comets  it  will  evi- 
dently be  necessary  to  adopt  different  intervals  in  different  parts  of 
the  orbit.  When  the  comet  is  in  the  neighborhood  of  its  perihelion, 
and  also  wlien  it  is  near  a  disturbing  planet,  the  interval  must  nocw- 
s  irily  be  much  smaller  than  when  it  is  in  more  remote  parts  of  its 
trbit  or  farther  from  the  disturbiTii»'  bodv. 

It  will  be  observed,  further,  that  since  the  double  integral  contains 
the  factor  o/,  if  we  multiply  the  computed  values  of  the  function  by 
cu',  this  factor  will  be  included  in  all  the  differences  and  sums,  and 
hence  it  will  not  appear  as  a  factor  in  the  formulae  of  integration. 
If,  however,  the  values  of  the  function  are  already  multiplied  by  w', 
and  only  tlie  single  integral  is  sought,  the  result  obtained  by  the 
formula  of  integration,  neglecting  the  factor  (or,  will  be  a)  times  the 
actual  integral  rccpiired,  and  it  must  be  divided  by  co  in  order  to 
obtain  the  final  result. 


164.  In  tlie  computation  of  the  perturbations  of  one  of  the  asteroid 
planet-s  for  a  period  of  two  or  three  years  it  will  rarely  be  necessary 
to  take  into  account  the  eflFect  of  the  terms  of  the  sei^ond  order  with 
respect  to  the  disturbing  force.  In  this  case  the  numerical  values  of 
the  expressions  for  the  forces  will  be  com])uted  by  using  the  valnes 
of  the  co-ordiilates  computed  from  the  osculating  elements  fur  tlie 
beginning  of  the  integration,  instead  of  the  actual  disturbed  values 
of  these  co-ordinates  as  recpiired  by  the  formulae  (8).  The  values  of 
the  second  differential  coefficients  of  ox,  o//,  and  dz  with  I'cspcct  to 
the  time,  will  be  determined  by  means  of  the  equations  (9).  H  the 
interval  <o  is  such  that  the  higher  orders  of  differences  may  be  neg- 
lected, the  values  of  the  forces  must  be  computed  for  the  successive 
dates  separated  by  the  interval  (o,  and  commencing  with  the  date 
'o  —  2*^''  corresponding  to  the  argument  a  —  w,  /„  being  the  date  to 
which  the  osc^liiting  elements  belong.     Then,  since  the  last  terms 


of  the  formulfB  for 


(Bx   d'<hj 


.  and 


iP'h 


df '  (W ' """  dt'  "^^'°^^^  ^'^>  ^y>  ^"^^  ^^>  ^^■''''''' 


are  the  quantities  .sought,  the  subsequent  determination  of  the  differ- 
ential coefficients  must  be  performed  by  successive  trials.  Since  the 
integral  must  in  each  case  be  equal  to  zero  for  the  date  t^,  it  will  be 
admissible  to  assume  first,  for  the  dates  t^  —  ^to  and  ^q  +  ^cj  corre- 
sponding to  the  arguments  a  —  (o  and  a,  that  8x  =-  0,  di/  —-  0,  and 
dz  =  0,  and  hence  that  the  three  differential   coefficients,  for  oaili 


VARIATIOX   OF   CO-ORDINATES. 


445 


date,  are  respectively  equal  to  A'^,,  }''„,  and  Z^.  We  may  now  by  inte- 
gration derive  the  aetual  or  tlu!  very  approximate  values  of  the 
variation.s  of  the  co-ordinates  for  the.^e  two  dates.  Thus,  in  the  ease 
of  each  co-ordinate,  we  compute  the  value  of  '/(«  —  Uo)  by  means 
of  the  equation  (26),  using  only  the  first  term,  and  the  value  of 
"J{a  —  to)  from  (36),  using  in  this  case  also  only  the  first  term.  The 
value  of  the  next  function  symbolized  by  "f  will  be  given  by 

7(a)  =:  "f(a  -  oO  +  '/(a  -  W). 

Then  the  formula  (39),  putting  first  /  —-  —  1  and  then  i  =  0,  and 
neglecting  second  differences,  will  give  the  values  of  the  variations 
of  the  co-ordinates  for  the  dates  a  —  o)  and  a.  These  operations  will 
1)0  performed  in  the  case  of  each  of  the  tlu'ee  co-ordinates;  and,  by 
moans  of  the  results.,  the  corrected  values  of  the  ditl'erential  coeffi- 
cients will  be  obiained  from  the  equations  (9),  the  value  of  or  being 
computed  by  means  of  (7).  With  the  corrected  values  thus  derived 
a  new  table  of  integration  will  be  commenced;  and  the  values  of 
'/(«  — l(o)  and  "f{a  —  (o)  will  also  be  recomputed.  Then  we  obtain, 
also,  by  adding  '/{a  —  |o;)  to  /(«),  the  value  of  '/(a  +  Uo),  and,  by 
adding  this  to  ''/(«),  the  value  of  "f(a  +  w). 

An  api)roximate  value  of /(a  +  to)  may  now  be  readily  estimated, 
and  two  terms  of  the  equation  (39),  putting  <==  1,  will  give  an  ap- 
proximate value  of  the  integral.  This  having  Ijeen  obtained  for 
each  of  the  co-ordinates,  the  corresponding  complete  values  of  the 
ditU'rential  coefficients  may  be  comj)uted,  and  these  having  been 
introduceil  into  the  table  of  integration,  the  process  may,  in  a  similar 
manner,  be  carried  one  step  farther,  so  as  to  determine  first  approxi- 
mate values  of  iXv,  (5//,  and  (h  for  the  date  represented  by  the  argu- 
ment a  H-  2(0,  and  then  the  corresponding  values  of  the  differential 
cootfieients.  We  may  thus  by  successive  partial  integrations  deter- 
mine the  values  of  the  unknown  quantities  near  enough  for  the  cid- 
ciilation  of  the  series  of  diffi'rential  coefficients,  even  when  the  inte- 
grals are  involved  directly  in  the  values  of  the  differential  coelfieients. 
If  it  be  found  that  the  assumed  value  of  the  function  is,  in  any  case, 
nnu'h  in  error,  a  repetition  of  the  cahndation  may  become  necessary ; 
Init  when  a  few  values  have  been  found,  the  course  of  the  function 
will  indicate  at  once  an  approximation  sufficiently  close,  since  what- 
ovor  error  remains  affects  the  aj)proximate  integral  by  only  one- 
twelfth  part  of  the  amount  of  this  error.  Further,  it  is  evident 
thai,  in  cases  of  this  kind,  when  the  determination  of  the  values  of 
the  dilferential  coefficients  requires  a  preliminary  approximate  into- 


446 


THEORETICAL   ASTROXOMY. 


gration,  it  is  necessary,  in  order  to  avoid  the  effect  of  tlie  errors  in 
the  values  of  the  liigher  orders  of  differences,  that  the  interval  (o 
should  he  smaller  than  when  the  successive  values  of  the  function  to 
be  integrated  arc  already  known.  In  the  case  of  the  small  planets 
an  interval  of  40  days  will  afford  the  re(|uired  facility  in  the  a])j)roxi- 
mations ;  but  in  the  ease  of  the  comets  it  may  often  be  necessary  to 
adopt  an  Interval  of  only  a  few  days.  The  necessity  of  a  change  in 
the  adopted  value  of  (o  will  be  indicated,  in  the  numerical  a[)pli(a- 
tion  of  the  formulte,  by  the  maimer  in  which  the  successive  assump- 
tions in  regard  to  the  value  of  the  function  are  found  to  agree  witli 
tlie  corrected  results. 

The  values  of  the  differential  coefficients,  and  hence  those  of  the 
integrals,  are  conveniently  expressed  by  adopting  for  unity  the  unit 
of  the  seventh  decimal  place  of  their  values  in  terms  of  the  unit  of 
space. 

165.  Whenever  it  is  considered  necessary  to  commence  to  take  into 
account  the  perturbations  due  to  the  second  and  higher  powers  of  the 
disturbing  force,  the  (  inplete  ecpiations  (14)  must  be  employed.  In 
this  case  the  forces  X,  Y,  and  Z  should  not  be  computed  at  once  for 
the  entire  period  during  which  the  perturbations  are  to  be  determined. 
The  values  computed  by  means  of  the  osculating  elements  will  lie 
employed  only  so  long  as  simply  the  first  power  of  the  disturl/ini; 
force  is  considered,  and  by  means  of  the  approximate  values  of  J,r, 
8i/,  and  fh  which  would  be  employed  in  computing,  for  the  next  plate, 
the  last  terms  of  the  equations  (9),  we  must  compute  also  the  cor- 
rected values  of  A',  Y,  and  Z.  These  will  be  given  by  the  second 
members  of  (8),  using  the  values  of  .v,  y,  and  z  obtained  from 


X  =  Xg  -{-  6x, 


y  =  yo  +  %, 


z,  +  oz. 


Wc  compute  also  q  from  (12),  and  then  from  Table  XVII.  find  the 

corresponding  value  of/.     The  corrected  values  o^  —A-,  -  ,','  and 

— .—  will  bo  given  by  the  equations  (14),  and  these  being  introduced, 

in  the  continuation  of  the  table  of  integration,  we  obtain  new  values 
of  ox,  01/,  and  8z  for  the  date  under  consideration.  If  these  ditl'er 
much  from  those  previously  assumed,  a  repetition  of  the  calculation 
will  be  necessary  in  order  to  secure  extreme  accuracy.  In  this  repe- 
tition, however,  it  will  not  be  necessary  to  recompute  the  coeificionts 
of  8x,  Sif,  and  8z  in  the  formula  for  q,  their  values  being  given  with 
sufficient  accuracy  by  means  of  the  previous  assumption ;  and  geue- 


VARIATION   OF   CO-ORDINATES. 


447 


^^TI.  fiiul  the 


rally  a  repetition  of  tlie  calculation  of  X,  Y,  and  Z  will  not  be 
reqiui'cd. 

Next,  the  values  of  ox,  6i/,  and  8z  may  be  determined  approxi- 
mately, as  already  explained,  for  the  followinj;  date,  and  by  means 
of  these  the  corresponding  values  of  the  forces  X,  V,  and  /  will  be 
found,  and  also/  and  the  remaining  terms  of  (14),  after  which  the 
integration  will  be  completed  and  a  new  trial  made,  if  it  be  con- 
sidered necessary.  In  the  final  integration,  all  the  terms  of  the  Ibr- 
mnho  of  integration  which  sensibly  aifect  the  result  may  be  taken 
into  account.  By  thus  performing  the  complete  calculation  of  each 
successive  })lace  separately,  the  determination  of  the  perturbations  in 
the  values  of  the  co-ordinates  may  be  cifcctcd  in  reference  to  all 
j)0\vers  of  the  masses,  provided  that  we  I'cgard  the  masses  and  co-or- 
dinates of  the  disturbing  bodies  as  being  accurately  known;  and  it  is 
api)arent  that  this  complete  solution  of  the  problem  re([uires  very 
little  more  labor  than  the  determination  of  the  pertnrl)ations  when 
only  the  first  power  of  the  disturbing  force  is  considered.  But 
altiiough  the  places  of  the  disturbing  bodies  as  given  by  the  tables 
of  their  motion  may  be  regarded  as  accurately  known,  there  are  yet 
the  errors  of  the  adopted  osculating  elements  of  the  disturbed  body 
to  detract  from  the  absolute  accuracy  of  the  computed  perturbations; 
and  hence  the  probable  errors  of  these  elements  should  be  constantly 
kept  in  view,  to  the  end  that  no  useless  extension  of  the  calculation 
may  be  undertaken.  When  the  osculating  elements  have  been  cor- 
rected by  means  of  a  very  extended  series  of  observations,  it  wili  be 
cx[)('dient  to  determine  the  perturbations  with  all  possible  rigor. 

When  there  arc  several  distiu'bing  planets,  the  forces  for  all  of 

these  may  be  computed  simultaneously  and  united  in  a  single  sum, 

so  that  in  the  equations  (14)  we  shall  have  IX,  ^'F,  and  -Z  instead 

of  A',  Y,  and  Z  respectively;  and  the  integration  of  the  expressions 

„     iP(Xv    (I'dy  il'ih      .,,     ,  .        ,  ,     .         ,  , 

tor     ,--.  ^T^T'  ^"^^  "7/F  ^^"*  u\G\\  give  the  perturbations  due  to  the 


iW 


<W 


(W 


action  of  all  the  disturbing  bodies  considered.  However,  when  the 
interval  co  for  the  different  disturbing  planets  may  be  taken  differently, 
it  may  be  considered  expedient  to  compute  the  perturbations  sepa- 
rately, and  especially  if  the  adopted  values  of  the  masses  of  some  of 
the  disturbing  bodies  are  regarded  as  uncertain,  and  it  is  desired  to 
separate  their  action  in  order  to  determine  the  probable  corrections 
to  be  applied  to  the  values  of  m,  m',  &i\,  or  to  determine  the  effect 
of  any  subsequent  change  in  these  values  without  repeating  the  cal- 
culation of  the  perturbations. 


448 


THEORETICAL   ASTRONOMY. 


166.  Example. — To  illustrate  the  numerical  application  of  tlio 
formuhc  lor  the  computation  of  the  pcrtui'bations  of  the  rectangular 
co-ordinates,  let  it  be  required  to  compute  the  perturbations  of 
Eurynomc  @  arising  from  the  action  of  Jnpiler  from  1864  Jan.  1.0 
Berlin  mean  time  to  1865  Jan.  15.0  Berlin  mean  time,  assuming  the 
osculatina:  elements  to  be  the  following : — 


i»/o  = 


Epoch  =  18G4  Jan.  1.0  J'jrliu  mean  time 
1°  29'  5".65 
17  12  .17 
39  5  .69 
36  52  .11 
15  51  .02 


^^0=    44 


fto  =  206 
4 
11 


Ecliptic  and  Mean 
Equinox  1860.0 


log  «o  =  0.3881319 
/i„=:928".55745 


From  these  elements  we  derive  the  following  values : — 


Berlin  Mean  Time. 

'h 

2/0 

% 

log  f'o 

1863  Dec. 

12.0 

+  1.53616 

+  1.23012 

—  0.03312 

0.294084, 

1864  Jan. 

21.0 

1.15097 

1.59918 

0.07369 

0.294887, 

IMarch 

1.0 

0.69518 

1.87033 

0.10978 

0.300(J74, 

April 

10.0 

+  0.19817 

2.03141 

0.13936 

0.  lO.sfU, 

^lay 

20.0 

—  0.31012 

2.08092 

0.161.34 

0.324298, 

June 

29.0 

0.80326 

2.02602 

0.17523 

0.33974.5, 

Aug. 

8.0 

1.26055 

1.87959 

0.18122 

0.35G101, 

Sept. 

17.0 

1.66729 

1.65711 

0.17990 

0.3724GI), 

Oct. 

27.0 

2.01414 

1.37473 

0.17209 

0.388214, 

Dec. 

6.0 

2.29597 

1.04766 

0.15870 

0.402894, 

1865  Jan. 

15.0 

—  2.51077 

+  0.68978 

—  0.14066 

0.416240. 

The  adopted  interval  is  (o  =  40  days,  and  the  co-ordinates  are  re- 
ferred to  the  ecliptic  and  mean  equinox  of  1860.0.  The  first  date, 
it  will  be  observed,  corresponds  to  t^  —  ^w,  and  the  integration  is  to 
commence  at  1864  Jan.  1.0. 

The  })laces  of  Jupiter  derived  from  the  tables  give  the  following 
values  of  the  co-ordinates  of  that  planet,  with  which  we  write  also 
the  distances  of  Eurynomc  from  Jupiter  computed  by  means  of  tlie 
formula 

P'  =  (.-«'  -  xf  +  {y'  -  yr  +  (/  -  z)\ 


Berlin  Mean  Time.  x' 

1863  Dec.      12.0  —4.09683 

1864  Jan.  21.0  3.89630 
March  1.0  3.68416 
April    10.0  —3.46098 


y 

-8.55184 
3.76053 
3.95803 

-4.14366 


+  0.10533 
0.10152 
0.09744 

+  0.09304 


log  r' 
0.73425 
0.73368 
0.73305 
0.73237 


log/' 
0.8G86G, 
0.86713, 
0.8G292, 
0.85G22, 


NUMERICAL   EXAMPLE. 


449 


:ion  of  tlie 
rectangular 
•bations  of 
1)4  Jan.  1.0 
snming  the 


log  I'o 

0.2940S4, 
0.294837, 
0.:5()0()74, 
0.  lO^H 
0.;i24298, 
0.339745, 
0.3.")()101, 
0.3724G9, 
0.3.S8214, 
0.402894, 
0.41()240. 

latos  arc  re- 
le  first  (lute, 
ffratioa  is  to 

le  following 
Ic  write  also 
Leans  of  the 


[25 
[68 
l05 

hi 


0.86866, 
0.86713, 
0.86292, 
0.85622, 


Berlin  Mean  Time. 

x' 

2/' 

s' 

logr' 

lORp 

1864  May  20.0 

—  3.22739 

—  4.31684 

+  0.08839 

0.73164 

0.84732, 

June  29.0 

2.98405 

4.47693 

0.08346 

0.73086 

0.83656, 

Aug.    8.0 

2.73162 

4.62343 

0.07827 

0.73003 

0.82428, 

Sept.  17.0 

2.47085 

4.75576 

0.07284 

0.72915 

0.81077, 

Oct.   27:0 

2.20247 

4.87345 

0.06720 

0.72823 

0.79628, 

Dec.     6.0 

1.92728 

4.97606 

0.06134 

0.72726 

0.78098, 

1865  Jan.   15.0 

—  1.64600 

—  5.06301 

+  0.05531 

0.72625 

0.76498. 

These  co-ordinates  are  also  referred  to  the  ecliptic  and  mean  equinox 
of  1860.0. 
If  we  neglect  the  mass  of  Earynome  and  adopt  for  the  mass  of 

Jupiter 


in  = 


1047.819' 


we  obtain,  in  units  of  the  seventh  decimal  place, 


m'lc'  =:  4518.27, 


aud  the  equations  (9)  become 
d'Sx 


to' 


dt' 


4518.27 


^^^  =  4518.27  (^- 


% 


d''h 
"dtf 


=  4518.27 


+ 


+ 


+ 


0.47346 


0.47346 

,,.3 
'0 

0.47346 


(40) 


Substituting  for  the  quantities 
of  each  of  these  equations  the 


in  the  first  terra  of  the  second  member 
values  already  found,  we  obtain 


Argument. 

Date. 

"'Xo 

'o'Yo 

OJ^Zo 

a  —  0) 

1863  Dec. 

12.0 

+  53.00 

+  47.09 

—  1.43, 

a 

1864  Jan. 

21.0 

53.71 

46.31 

0.91, 

a  -\-  10 

March 

1.0 

54.23 

45.18 

—  0.37, 

a  +  2m 

April 

10.0 

54.69 

43.59 

+  0.22, 

«  +  3w 

May 

20.0 

55.23 

41.51 

0.70, 

a  -}-  4w 

June 

29.0 

56.06 

38.96 

1.19, 

a  +  5'" 

Aug. 

8.0 

57.30 

35.92 

1.66, 

U  +  Gta 

Sept. 

17.0 

59.09 

32.47 

2.08, 

a-\-7<o 

Oct. 

27.0 

61.55 

28.60) 

2.43, 

a +  8(0 

Dec. 

6.0 

64.85 

24.34 

2.69, 

a  -f  9w 

1865  Jan. 

15.0 

+  69.09 

+  19.78 

+  2.83, 

which  are  expressed  in  units  of  the  seventh  decimal  place. 
Wo  now,  for  a  first  approximation,  regard  the  perturbations  as 


29 


450 


THEORETICAL   ASTIIONOMY. 


being  equal  to  zero  for  the  dates  Dec.  12.0  and  Jan.  21.0,  and,  in  the 
case  of  the  variation  of  x,  we  compnte  first 


"/(a-io) 


2',/'  (a  —  }.oj)^-  2',  (53.71  —  53.00)  =  -  0.03, 
53.71 


3'r/(«)  =  + 


24 


-  +  2.24, 


and  the  api)roximate  table  of  integration  becomes 


f(a  —  w)  =  +  .53.00  , 


-nn-^  "/(«-*")  =  +  2.24, 


/(«)  =  +  53.7l'^^"-^->=^-^-^^7(«)         =  +  2.21. 

Then  the  formula  (39),  putting  first  i  =  —  1,  and  then  i  =  0,  gives 


Dec.  12.0 
Jan.  21.0 


SX: 


+  2.24  +  ^-^^ 


+  2.21  + 


53.7^^ 
12 


+  6.66, 
+  6.69. 


In  a  similar  manner,  we  find 


Dec.  12.0 
Jan.  21.0 


Sy  =-.  4-  5.85 
dy  =  +  5.82 


dz  =  —  0.16, 
fe  =  —  0.14. 


By  means  of  these  results  we  compute  the  complete  values  of  the 
second  members  of  equations  (40),  8r  being  found  from 


and  thus  we  obtain 
Date. 

Dec.  12.0 
Jan.  21.0 


dr  =  ^Sx  +  y-^Sy-\-^^3z, 


U'  ■■ 

+  53.86 
+  54.23 


6)5 i 

-f  47.76 
+  47.25 


1.45 
0.96 


6r 

+  8.85, 
+  8.63. 


We  now  commence  anew  the  table  of  integration,  namely. 


'/       7 


y 


7 


'/      "/ 


+53.86 _  ..^+  2.26,  +47.76  ,    ..2+  1-97,  -1-45     n 02 -'^•^^' 
+54.23  ,  5^-21  +  2.24,  +A7.25'\_^^[^,^  ^  1.99,  -0.96  _Q;9g -0.06, 


+56.45, 


+49.26, 


-1.04, 


the  formation  of  which  is  made  evident  by  what  precedes. 

Wc  may  next  assume  for  approximate  values  of  the  differential 
coefficients,  for  the  date  March  1.0,  +  54.6,  +  46.7,  and  —  0.5, 
respectively ;  and  these  give,  for  this  date, 


NUMERICAL   EXAJfPLE. 


451 


<J*  =  +  56.45  +  ^  =  +  61.00, 
ay  =  + 49.26 +  -j-^  =  + 53.15, 


5z  =  —   1.04  — 


a5_ 

12 


=  —    1.08. 


By  means  of  these  ai)i)roximate  values  we  obiain  the   following 
results : — 

+  55.01.    ^if  =  + 53.86.      «  = 
Jr  =  +  71.03. 


1804  March  1.0     ^'^J 


1.00, 


Introducing  these  into  the  table  of  integration,  we  find,  for  the  corre- 
sponding values  of  the  integrals, 


<5a;  =  -f- 61.03, 


5?,  =  4-  53.75, 


52=  —  1.12. 


Those  results  differ  so  little  from  those  already  derived  from  the 
assumed  values  of  the  function  that  a  repetition  of  the  calculation  is 
unnecessary.     This  repetition,  however,  gives 


=  +  55.04, 


^  =  -x.oa 


Assuming,  again,  approximate  values  of  the  differential  coefficients 
for  April  10.0,  and  computing  the  corresponding  values  of  ox,  oy, 
and  8z,  we  derive,  for  this  date, 


,..'^=  +  «.oe, 


df 


+  63.19,        a.^^^-  =  -lM. 


Introducing  these  into  the  table  of  integration,  and  thus  deriving 
approximate  values  of  ox,  dy,  and  dz  for  May  20,  we  carry  the  pro- 
cess one  step  further.  In  this  manner,  by  successive  approximations, 
wc  obtain  the  following  results : — 


Date. 


,dMj; 


,rf% 


,<P<h 


x/aic. 

"  df^ 

-  dfl 

dt^ 

1863  Dec. 

12.0 

+  53.86 

+  47.76 

-1.45, 

1864  Jan. 

21.0 

54.23 

47.25 

0.96, 

Marcli 

I    1.0 

55.04 

53.91 

1.00, 

April 

10.0 

48.06 

63.19 

1.54, 

May 

20.0 

32.85 

65.40 

2.07, 

June 

29.0 

16.74 

64.48 

1.75, 

Aug. 

8.0 

8.62 

31.39 

—  0.36, 

Sept. 

17.0 

+  14.20 

+    2.09 

+  1.86, 

452 


THEORETICAL  ASTRONOMY. 


Date. 

Mr. 

c/> 

(If 

-7 

df 

1864  Oct. 

27.0 

+    34.84 

—  26.32 

+  -1.44, 

Dec. 

6.0 

68.79 

47.87 

6.86, 

1865  Jan. 

15.0 

+  112.64 

—  58.39 

-t-  8.68. 

The  complete  integration  may  now  be  effected,  and  we  may  use  both 
equation  (37)  and  equation  (39),  the  former  giving  the  integral  for 
the  dates  Jan.  1.0,  Feb.  10.0,  March  21.0,  &g.,  and  tlie  latter  the 
integrals  for  the  dates  in  the  foregoing  table  of  values  of  the  function. 
The  final  results  for  the  perturbations  of  the  rectangular  co-ordinates, 
ex])ressed  in  units  of  the  seventh  decimal  place,  are  thus  found  to  be 
the  following: — 


Berlin  Mean  Time. 

6x 

<V 

in 

1863  Dec. 

12.0 

+  6.7 

+  5.9 

-0.2, 

1864  Jan. 

1.0 

0.0 

0.0 

0.0, 

21.0 

+  6.8 

8.9 

0.1, 

Feb. 

10.0 

27.1 

23.5 

0". 

March    1.0 

61.0 

53.7 

1.1, 

21.0 

108.9 

97.4 

2.0. 

April 

10.0 

169.7 

155.7 

3.1, 

30.0 

242.7 

229.9 

4.7, 

May 

20.0 

325.7 

320.3 

6.7, 

June 

9.0 

417.1 

427.2 

9.3, 

29.0 

614.6 

549.1 

12.3, 

July 

19.0 

616.1 

684.9 

15.7, 

Aug. 

8.0 

720.8 

831.4 

19.5, 

28.0 

827.4 

986.0 

23.4, 

Sept. 

17.0 

936.8 

1144.6 

27.0, 

Oct. 

7.0 

1049.4 

1303.8 

30.2, 

27.0 

1168.2 

1460.0 

32.6, 

Nov. 

16.0 

1295.4 

1609.4 

33.9, 

Dec. 

6.0 

1435.6 

1749.6 

33.8, 

26.0 

1592.8 

1877.6 

32.0, 

1865  Jan. 

15.0 

+  1772.6 

+  1992.3 

—  28.2. 

During  the  interval  included  by  these  perturbations,  the  terms  of 
the  second  order  of  the  disturbing  forces  will  have  no  sensible  effect; 
but  to  illustrate  the  application  of  the  rigorous  formulae,  let  us  com- 
mence at  the  date  1864  Sept.  17.0  to  consider  the  pertui'bations  of 
the  second  order. 

In  the  first  place,  the  components  of  the  disturbing  force  must  be 
computed  by  means  of  the  equations 


NUMERICAL   EXAMPLE. 


453 


The  approximate  valuoM  of  dv,  ni/,  and  dz  for  Sept.  17.0  given  imme- 
diately by  the  table  of  integration  extended  to  this  date,  will  .snfHce 
to  furnish  the  required  values  of  the  disturbed  co-ordinates  by  uieans 

of 

and  to  find  /o  =  jf'o  +  ^P)  ^^^  '^'^ve 

,  X' X  ,  V  —  W  .  sf  —  3  , 

5p  = —  dx  — Sy Sz, 


or 


5  log  />  -  -  4  ((x'  -  x)  Sx+OJ-  y)  dy  +  (z'  -  z)  'h), 

in  Avhioh  ^^  is  the  modulus  of  the  system  of  logarithms.     Thus  we 
obtain,  for  Sept.  17.0, 


<j^X=  +  59.09, 


,Jlog|0=  + 0.0000084, 


'Z  =-1-2.08, 


which  requii'e  no  further  correction. 
Next,  we  compute  the  values  of 


X,  +  ^8x 


Vo  +  3'^.y 


2o+A'53 


V  V  'V 


which  also  will  not  require  any  further  correction,  and  thus  we  form, 
according  to  (12),  the  equation 

5  =  —  0.29996o"a;  +  0.2QS15Sy  —  0.03237o2. 

The  approximate  values  of  dx,  dy,  and  dz  being  substituted  in  this 

equation,  we  obtain 

5  —  -f.  0.0000061, 

corresponding  to  which  Table  XVII.  gives 

log/=  0.477115. 


Hence  we  derive 

(lyU 


ID 


'■¥ 


(fqx  -  dx)  =  -  44.87,        — ,-  (fqy  ~  Sy)  =  -  30.40, 


--^(/32-<J2)  =  -0.21, 

'o 


4o4 


THEORETICAL  ASTRONOMY. 


uiul  the  ofjuations  (M)  give 


dt 


:  -  +  14.22, 


^^^^^  +  2.08, 


(P'h 
dt' 


+  1.87. 


These  values  being  introduced  into  the  table  of  int(>gration,  the 
resulting  values  of  the  integrals  are  changed  so  little  that  a  repetition 
of  the  ealculation  is  not  required. 

We  now  derive  approximate  values  of  dv,  (\i/,  and  dz  for  Oct.  27.0, 
and  in  a  similar  manner  we  obtain  the  (jorrected  values  of  the  dilfoi- 
ential  coefficients  ibr  this  date;  and  thus  by  computing  the  forces  for 
each  pla(^e  in  succession  from  approximate  values  of  the  perturbations, 
and  repeating  the  calculation  whenever  it  may  appear  necessary,  wc 
may  determine  the  perturbations  rigorously  for  all  powers  of  the 
masses.  Tiie  results  in  the  case  under  consideration  are  the  follow- 
ing:- 


Date. 

dt'' 

'''  di^ 

1864  Sept.  17.0 

+  14.22 

+  2.08 

+  1.87, 

Oct.  27.0 

34.84 

-  26.31 

4.44, 

Dec.  6.0 

68.77 

47.86 

0.86, 

1865  Jan.  15.0 

+  112.60 

-  .58.39 

+  8.68. 

Introducing  these  results  into  the  table  of  integration,  the  integrals 
for  Jan.  15.0  are  found  to  be 

to  =  +  1772.6,        ^2/  =  +  1992.3,        ^s  =  —  28.2, 

agreeing  exactly  with  those  obtained  when  terras  of  the  order  of  the 
square  of  the  distiu'bing  forces  are  neglected. 

If  the  perturbations  of  the  rectangular  co-ordinates  referred  to  the 
equator  are  required,  we  have,  whatever  may  be  the  magnitude  of  the 

perturbations, 

dx,  =  dx, 

dy,  =  cos  £  (hj  —  sin  e  Sz,  (41) 

Sz,  ==:  sin  £  Sy  -\-  C09  £  dz, 

x„  y„  z,  being  the  co-ordinates  in  reference  to  the  equator  as  the  fun- 
damental plane.     Thus  we  obtain,  for  1865  Jan.  15.0, 

0%  =  +  1772.6,        dy,  -=  +  1838.9,        8z,  =  -|-  767.2. 

These  values,  expressed  in  seconds  of  arc  of  a  circle  whose  radius  i;- 
the  unit  of  space,  are 

dx,  =  -f  36".562,        Sy,  =  -f  37".930,        Sz,  =  -f- 15".825. 


VAUIATION   OK  CO-ORDINATES. 


4oo 


The  approximate  geocentric  place  of  the  planet  for  the  .same  date  is 

a  =  183°  28',        3  =  —  f)"  ;?9',  log  A  =  0.:522!), 

luul  hence,  neglecting  terms  of  the  second  order,  we  derive,  Ity  means 
of  the  e(juations  (.'])..,  for  the  pertnrhations  of  the  geocentric  right 
ascension  and  declination, 

Aa  .-==  —  IT'M,  A')  =  +  r,".Cy7. 

I(i7.  The  values  of  dx,  oi/,  and  oz,  com[)uted  hy  means  of  the  co- 
ordinates referred  t(»  tiie  ecli[)tic  and  mean  e(ininox  of  the  date  /,  must 
ho  added  to  the  co-ordinates  given  by  the  undistnrl)e<l  elements  and 
referred  to  the  same  mean  equinox.  The  co-ordinates  referred  to  the 
eelii)tic  and  mean  e(piinox  of  t  may  he  readily  transfornuMl  into  those 
referred  to  the  ecliptic  and  mean  eijuinox  of  another  date  t'.  Thus, 
let  0  denote  the  longitude  of  the  descending  node  of  the  ecliptic  of  t' 
on  that  of  t,  measured  from  the  mean  e(|uinox  of  /,  and  let  3y  he  the 
mutual  inclination  of  these  planes;  then,  if  we  denote  hy  x',  )/',  z' 
the  co-ordinates  referred  to  the  ecliptic  of  t  as  the  fundamental  plane, 
the  positive  axis  of  x,  however,  being  directed  to  the  point  wdiose 
longitude  is  d,  wo  shall  have 


x'  1=  .c  cos  fl  +  y  ^'w  "i 
l/  =  —  X  sin  0  -\-  y  cos  0, 


(42) 


Let  us  now  denote  by  x",  }/",  z"  the  co-ordinates  when  the  ecliptic 
of  i  is  the  plane  of  .r^,  the  axis  of  x  remaining  the  same  as  in  the 
system  of  x' ,  y',  z'.     Then  we  shall  have 


a;"  =  X, 

y"  =  7/'  cos  TJ  —  z  sin  rj, 

a"  =  y'  sin  Tj  -\-  z'  cosi?. 


(43) 


Finally,  transforming  these  so  that  the  axis  of  ;:  remains  unchanged 
while  the  positive  axis  of  x  is  directed  to  the  mean  ecpiinox  of  t,  and 
denoting  the  new  co-ordinates  by  x„  i/,,  s„  we  get 


X,  =  x"  cos  {0  -f  J))  —  y"  sin  (0  +  j)), 
y,  =  x"  sin  (tf  +  i>)  -f-  y"  cos  {0  +  p), 


(44) 


z,  =z  , 


in  which  p  denotes  the  precession  during  the  interval  t'  —  t.  Elimi- 
nating x",  y",  and  z"  from  these  ecpiations  by  means  of  (43)  and  (42), 
observing  that,  since  ;y  is  very  small,  we  may  put  cos;y  =  1,  we  get 


456 


THEORETICAL   ASTllOXOMY. 


7 

X,  --■-.  X  iMsp  —  ij  sriiy>  --(-  -  2  siu  (0  -\- 2>), 


'I 
y,  —-.  .r  siu  p  -\-  y  coi*p  —  ~z  cos  {0  ■■{-  j)), 

s 

Ti  .  JJ 

z,  T=zz  —  -  X  SIU  0  4-  ••  '/  <'0S  0, 

tf  ,1  ' 


(4o:t 


in  which  .s    -  2O(J204.<S,  r^  bciiifij  supposed  to  be  expressed  in  secoiuls 
of  arc.      If  we  neglect  terinn  of  the  order  j/,  these  e([uations  become 


X,  =:;  X  —  V-j,  ./•  —  1  //  4-  _  (sin  <?  -f  p  COS  0)  z, 

y>^y  -~  h  ^7  y  +  ~  -^  —  7  C^os  o—p  sin  <?)  3, 

1^  »s  «*• 

*?       •  ■*? 

z,  =^z  —    X  sni  fl  -|-  -  y  cos  ^. 


(46) 


These  forninlte  j;ive  the  (!o-ordi nates  referred  to  the  ecliptic  and  mean 
equinox  of  ojie  cixkjIi  when  those  referred  to  the  eeliptii-  and  mean 
e<{ninox  of  another  date  are  kuo\vn.  For  the  values  ol' 7>,  ^,  and  f', 
we  hav(! 

■p  =  (50".21120  -f  0".00n24429(!0r")  {t'  —  0, 

Tj  n=(  0",488<)2  — 0".()<)0()()(q43T)  (t'-~t), 

0  r.r.-  351°  ;Ui'  10"  -f  3;)".7i>  a  -  IToO)  -  5".21  (1!  —  t), 

ill  which  c  ---■■  l{t.'  --  t)  —  1750,  t  and  t'  beint^  expressed  in  years  from 
the  beginnine;  of  the  era.  If  we  add  the  nutation  to  the  value  of  p, 
the  co-ordinates  will  be  derived  for  the  true  equinox  of  f. 

The  e((uations  (45)  and  (40)  serve  also  to  convert  the  values  of  ox, 
3>f,  and  Sz  belonging  to  the  co-ordinates  referred  to  the  eclij)ric  and 
mean  equinox  of  f  into  tliose  to  be  a])])lied  to  the  co-(n'diiiates  re- 
ferred to  the  ecli])tie  and  mean  equinox  of  t\  For  this  purpose  it 
is  only  necessary  to  write  d.r.  dy,  and  oz  in  place  of  .c,  y,  and  -  re- 
spectively, and  similarly  for  .i-,,  y„  z,. 

In  the  eoinputation  of  the  perturbations  of  a  heavenly  b(jdy  ditriii[r 
a  period  of  several  years,  it  will  be  ;,-  (Venient  to  adopt  a  tixed  ecpii- 
nox  and  ecliptic  throughout  the  calculation;  but  when  the  pcrturlni- 
tions  a''c  to  hv  a})plie(l  to  the  co-ordinates,  in  the  calculation  of  an 
epheineris  of  the  body  taking  into  account  the  [)erturl)atioi  s,  it  will 
be  convenient  to  comjmte  the  co-ordinates  directly  for  the  cclijitic 
and  mean  e(iuinox  of  the  beginning  of  the  year  for  which  tiio 
ephenieris  is  re<|uired,  and  the  values  of  ox,  dy,  and  dz  nnist  he 
reduced,  by  means  of  tin;  equations  (45),  as  already  ex[)lained,  fiom 
the  ecli})tic  and  mean  equinox  to  which  they  belong,  to  the  ecliplii; 
and  mean  equinox  adopted  in  the  ease  of  the  co-ordinates  required. 


VARIATION  OF   CO-ORDIXATES. 


457 


In  a  similar  manner  wo  may  derive  Ibrmuliv  lor  the  transformation 
of  the  co-ordinates  or  of  their  variations  referred  to  the  mean  e([ninox 
and  equator  of  one  date  into  those  referred  to  the  mean  equinox 
;uid  equator  of  another  date;  hut  a  transformation  of  this  kind  will 
rarely  bo  refjuired,  and,  wlienevor  required,  it  may  be  etfeeted  by  lirst 
ciinvertinii;  the  co-ordinates  relerred  to  the  equator  into  those  referred 
to  the  ecliptic,  reducing  these  to  the  equinox  of  t'  by  means  of  (45) 
or  (46),  and  tinally  converting  them  into  the  values  referred  to  tho 
equator  of  t' .  Since,  in  tho  conij)utation  of  an  e]>hemeris  for  the 
comparison  of  observations,  tl)e  co-ordinates  are  i'"nerally  required 
ill  reference  to  the  equator  as  the  fundamental  plan<-.  it  would  ai)i)ear 
preferable  to  ado[)t  this  plane  as  the  plane  of  xy  in  the  computation 
of  the  perturbations,  and  in  some  cases  this  method  is  most  advan- 
tageous. But,  generally,  since  the  elements  of  the  orbit  of  the  dis- 
turbed planet  as  well  as  the  elements  of  the  orbits  of  the  disturbing 
bodies  are  referred  to  the  ecliptic,  the  calculation  of  the  perturbations 
vrill  be  most  conveniently  perlbrmed  by  adopting  the  (vliptic  as  tue 
fundamental  plane.  The  consideration  of  the  change  of  the  position 
of  the  fundamental  plane  from  one  epoch  to  another  is  thus  also  ren- 
dered more  simple.  Whenever  an  ephemeris  giving  the  geocentric 
right  ascension  and  declination  is  required,  the  heliocentric  co-ordi- 
nates of  the  body  referred  to  the  mean  equinox  and  ecpiator  of  the 
l)o<j;inuin<x  of  the  vear  will  be  comi)uto<l  bv  means  of  the  osculatinsc 
elements  corrected  for  precession  to  that  epoch,  and  the  j)erturbations 
of  the  co-ordinates  referred  to  the  ecliptic  and  mean  etjuinox  of  any 
other  date  will  be  iirst  corrected  according  to  the  equations  (46),  and 
then  converted  into  those  to  be  applied  to  the  co-ordinates  referred  to 
the  mean  equinox  and  c<pi:itor.  If  the  perturbations  are  not  of  con- 
.■^ideraltle  ma<>'nitude  and  tlie  interval  t'  —  t  h  also  not  verv  larw,  the 
con'ection  of  ox,  d;/,  and  o-  on  account  of  the  change  of  the  j)osition 
of  the  ecliptic  and  of  the  equinox  will  be  insignificant;  and  the 
conversion  of  the  values  of  these  (piantities  referred  to  the  ecliptic 
into  the  corresponding  values  for  the  equator,  is  cflfected  with  great 
ficility. 

In  the  determination  of  the  perturbations  of  comets,  c|»hemerides 
being  required  only  tluring  the  time  of  describing  a  small  ])()rti(in  of 
their  orbits,  it  will  sometimes  be  convenient  to  adopt  th(>  plane  of  the 
uiulisturbed  orbit  as  the  fundamental  plane.  In  this  ease  the  posi- 
tive axis  of  X  should  be  directed  to  tho  ascending  node  of  this  plane 
on  the  ecliptic,  and  the  subsequent  change  to  the  ecliptic  and  ecpiinox, 
whenever  it  may  be  required,  will  be  readily  etfeeted. 


I 


458 


THEORETICAL   ASTRONOMY. 


1G8.  The  ])crtiu'l)ations  of  a  licuvcnly  l)o(ly  may  thus  bo  deter- 
mined vigorously  for  a  long  period  of  time,  ])rovided  that  the  oscu- 
lating elements  may  be  regarded  as  accurately  known.  The  peculiar 
object,  however,  of  such  calculations  is  to  facilitate  the  correction  of 
the  assumed  elements  of  the  orbit  by  means  of  additional  observa- 
tions according  to  the  methods  which  have  already  been  explained; 
and  when  the  osculating  elements  have,  by  successive  corrections, 
been  determined  with  great  })recision,  a  repetition  of  the  cali'ulation 
of  the  j)erturbations  may  become  necessary,  since  changes  of  the  ele- 
ments which  do  not  sensibly  aifect  the  residuals  for  the  given  diifer- 
ential  equations  in  the  determination  of  the  most  probable  corrections, 
may  have  a  much  greater  influence  on  the  accuracy  of  the  resulting 
values  of  the  perturbations. 

^^^len  the  calculation  of  the  perturbations  is  carried  forward  for  a 
long  period,  using  constantly  the  same  osculating  elements, — and 
those  which  are  supposed  to  icr|uire  no  correction, — the  secular  per- 
turbations of  the  co-ordinates  arising  from  the  secular  variation  of 
the  elements,  and  the  perturbati<jns  of  long  period,  will  constantly 
affect  the  magnitude  of  the  resulting  values,  so  that  f).r,  oi/,  and  oz 
>vill  not  again  become  sinudtaneously  equal  to  zero.  Ilenee  it 
appears  that  even  when  the  ado])ted  elements  do  not  diifer  much 
from  their  mean  values,  the  numerical  amount  of  the  perturbations 
may  be  very  greatly  inex'eased  by  the  secular  perturbations  and  by 
the  large  perturbations  of  long  period.     But  when  the  perturbations 

are  large,  the  calculation  or  the  complete  values  oi  —jfT>     lii  >  '^"'^' 

—777-  (which  is  eifected  indirectly)  cannot  be  performed  with  facility, 

rc|uiring  often  several  repetitions  in  order  to  obtain  the  retfuirod 
accuracy,  since  any  error  in  the  value  of  the  second  differential  coeffi- 
cient produces,  by  the  double  integration,  an  error  increasing  propor- 
tionally to  the  time  in  the  values  of  the  integral.  Errors,  therefore, 
in  the  values  of  the  second  differential  coefficients  which  for  a  modo- 
rate  period  would  have  no  sensible  effect,  may  in  the  course  of  a  long 
])eriod  produce  large  errors  in  the  values  of  the  perturbations,  and  it 
is  evident  that,  both  for  convenience  in  the  numerical  calculation  and 
for  avoiding  the  accumulation  of  error,  it  will  be  necessaiy  from  time 
to  time  to  apply  the  perturbations  to  the  elements  in  order  that  the 
integrals  may,  in  the  case  of  each  of  the  co-ordinates,  be  again  e(|iial 
to  zero.  The  calculation  will  then  be  continued  until  another  chan«;o 
of  the  ck«»ents  is  required. 


CHANGE   OF   THE   OSCULATING    ELEMENTS. 


459 


Tlie  transforiaation  from  a  system  of  osculating  elements  for  one 
epoch  to  that  for  another  epoch  is  very  onsilv  cfrected  by  means  of 
the  values  of  tlie  i)erturbations  of  the  eo-onlinates  in  connection 
with  the  corresponding  values  of  the  variations  of  the  velocities 

dx     dy  ,    rfs  rni        1 

rfl'  'dt'  di'  ^^^^^'^"  obtained  from  i\w.  values  of  the 

second  differential  coefficients  by  nieans  of  a  sin<j;le  intejrration  ac- 
cording  to  the  equations  (27)  and  (;32).  Thus,  in  the  case  of  the 
oxamjile  given,  we  obtain  for  the  date  18C5  Jan.  15.0,  by  means  of 
(32),  in  units  of  the  soyeuth  decimal  pla(  3, 


40f  =  +  385.9, 


40  "^^  =  +  214.6,        40 


toz 


+  9.7. 


The  velocities  in  the  case  of  the  disturbed  orbit  will  be  given  by  the 
formula) 

dx  __(b^      dtU  dj  _  f?i/o       d'hi  dz^  _  dz^       doz 

<••/"       dt^'dt'         dt~'dt'^dt'         dt'-'dt^^-     '^^'^ 


To  obtain  the  expressions  for  the  components  of  the  velocity 
resolved  parallel  to  the  co-ordinates,  we  have,  according  to  the  equa- 
tions (6),, 

dx         .        .    ,  ,    ,     ^  ^r  dv 

—  sm  a  sm  {A  +  u)     ,  +  r  sm  a  cos  {A  +  u)  -y-, 
fit  dt 


dt 

^  =  sin  c  sin  ( C  +  u)  ''-  +  r  sin c  cos  {C  +  u) -^ 


=  sin  i  sin  (5  4  u)  -jr  +  r  sin  h  cos  (B  +  u)  4-, 
«'  dt 


dt 


dt 


These  equations  are  applica'oie  in  the  case  of  any  fundamental  plane, 
if  the  auxiliaries  sin  «,  sin />,  ^in  ^,  A,  /i,  and  Care  determined  in 
reterence  to  that  plane.     To  transform  them  still  further,  we  have 


dr       kV'l  +  m      .    , 

-— . -_^  -     ^-= enm'u  —  w), 


Vp 


dp     Jcyp{i-{m.)  _  kVl  f 


'^  dt"" 


m 


/- — (1  4  e  cos  («  —  io)V 
Vp 


in  whk4»  lo  denotes  the  angular  dLstaiiee  of  the  perihelion  from  the 
ascending  iKxle.     Substit  ■  ing  i\tt?m  .aiues,  we  obtain,  by  reduction, 


460 


THEORETICAL   ASTRONOMY, 


(Ix 
'dt 

dy 
dt 

(h 
dt 


-jl ((e  cos  w  -f-  cos  u)  cos,  A  —  (e  sin  w  -f-  sin  u)  sin  A)  sin  u, 

Vp 


kV^  i'-\-7n_ 
Vp 

Vp 
Let  us  now  put 


((e  cos  u)  -{-  cos  It)  cos  B  —  (e  sin  w  -|-  sin  «)  sin  JS)  sin  b, 
((e  cos  w  +  cos  u)  cos  C  —  (e  sin  w  -f  sin  u)  sin  C)  sin  c. 


m 


and  we  have 


kVi  + 
Vp 

kVl  +  »t 

rf.r  _ 
rf?/  _ 

lit  ~ 

dz 
'dt' 


(e  sin  w  +  sin  u)  =  Fsin  C7, 
(e  cosw  -f-  cosft)  =  Fcos  U, 

=  Fsin  a  cos  (J.  -|-  U), 
=  Fsin  b  cos  (B  +  U), 
=  Fsinccos(C'4-  U). 


(48) 


(49) 


These  equations  determine  the  components  of  the  velocity  of  a  hea- 
venly body  resolved  in  directions  parallel  to  the  co-ordinate  axes, 
and  for  any  fundamental  plane  to  which  the  auxiliaries  A,  B,  etc. 
belong.     When  the  ecliptic  is  the  fundamental  plane,  we  have 

sin  c  =  sin  /,  C  =  0. 

The  sum  of  the  squares  of  the  equations  (48)  gives 

y2  ^  ^i(l+!!L)  (1  +  e^  +  2e  cos  («  -  w))  --.  /t»(l  -f.  m)  (  ?  -  -  \ 
p  \t      a) 

and  hence  it  appears  that  I'is  tho  linear  velocity  of  the  body. 

Tlie  dctonnination  of  the  osculating  "Icments  corresponding  to  any 
date  for  which  the  perturbations  of  the  co-ordinates  and  of  the  veloci- 
ties have  been  found,  is  thorcfvire  efl"  «'ted  in  the  folloviug  mat>rior: — 

First,  by  means  of  the  osculating  elements  to  which  the  jvrturl)a- 
tions  belong,  we  compute  accur.ii     values  of  )*q,   r:„,  ?/„,  5,.,  mid  l)y 

means  of  the  equations  (4H)  and  (49)  we  compute  th*-  vaJu*^  of 
-rn  and  -jt"     Then  we  apply  to  these  the  valw*^  of  the  perturba- 
tions, and  thus  find  .r,  ]j,  z,  -.-,  J ,  and  -  -•    Thest-   having  been 


(b: 
dt' 


dV  dt' 


dt 


CIIANGK   OF   TIIK   OSCULATING    ELEMENTS. 


461 


ibuiul,  the  equations  (32)j  will  furnish  the  values  of  SI,  i,  fvnti  p; 
and  the  remaining  elements  may  be  cletermined  as  explained  in  Art. 
112.     Thus,  li-om 


Fr  sin  4o  =  H>{l  +  m), 
Frcos+„  =  ..^^^+^--  +  .-^-, 

we  obtain  T';-  and  i^/dj  '-^^^^  from 

?•  sin  ?t  =  ( —  X  sin  Q,  -\-  y  cos  S^)  sec  i, 
r  cos  it  :=  X  cos  S^  +  2/  si  i  Q, , 

we  derive  r  and  «t;  and  hence  T^frona  the  value  of  Vr.     When  /  is 

not  very  small,  we  may  use,  instead  of  the  preceding  expression  for 

'/•sin  u, 

V  sin  n  =  3  cosec  i. 

Next,  we  compute  a  from 

2a  —  r 


and  from 


2  ^ni+j'0_i 

2ae  sin  w  =  —  (2a  —  7')  sin  (24.^  -{-  '0  —  ''  sin  «, 
2aB  cos  w  =  —  (2a  —  ?•)  cos  ( 24^  +  «)  —  r  cos  «, 


we  find  la  and  c.     The  mean  daily  motion  and  the  mean  anomaly  or 
tlie  mean  longitude  for  the  epoch  will  then  be  determined  by  means 
of  the  usual  fbrmuUe. 
In  the  case  of  a  very  eccentric  orbit,  after  r  and  n  have  been  found, 

-r  will  be  given  by  equations  (48)5,  and  the  values  of  c  and  v  will 

be  given  by  the  equations  (49)^.     Then  the  perihelion  distance  will 

be  found  from 

V 
■"       1  +  e 

and  the  time  of  perihelion  passage  will  be  found  from  v  and  c  by 
means  of  Table  IX.  or  Table  X. 

In  the  numerical  values  of  the  velocities  —^r,  — ,-,  etc.,  more  decimals 

at     at         ' 

must  be  retained  than  in  the  values  of  the  co-ordinates,  and  enough 

must  be  retained  to  secure  the  required  accuracy  of  the  sijlution.     If 

it  be  considered  necessary,  the  different  parts  of  the  calculation  may 

1)0  checked  by  means  of  various  formula;  which  liave  already  been 

Srivon.     Thus,  the  values  of  SI  and  i  must  satisfy  the  equation 


We  have,  also, 


462  THEORETICAL   AHTRONO^rY. 

3  COS  i  —  y  sill  t  COS  Q  -f-  .v  sin  /  sin  JJ  =  0. 

r'  :=  x'^  -\-  if  -\-  z\ 
z  =  r  sin  u  sin  i, 

■\vliich  must  be  satisfied  by  the  resulting  values  of  V,  r,  aud  u;  and 
the  values  of  a  aud  e  must  satisfy  the  equation 

ji  =  a(l  —  e'')  =  a  cos"  f. 

169.  When  the  plane  of  the  undisturbed  orl)it  is  adopted  as  the 
fundamental  plane,  we  obtain  at  once  the  i)erturbations 


iJ(?'C0Slt)) 


S  (r  sin  n), 


Sz, 


and  from  these  the  perturbations  of  the  polar  co-ordinates  are  easily 
derived.  There  are,  however,  advantages  which  may  be  secured  by 
employing  fornudai  Avliieh  give  the  perturbations  of  the  polar  co-or- 
dinates directly,  retaining  the  plane  of  the  orbit  for  the  date  t^  as  the 
fundamental  jdanc. 

Let  IV  denote  the  angle  which  the  projection  of  the  disturbed 
radius-vrctor  on  the  plane  of  xy  makes  with  the  axis  of  x,  arid  /9  tlie 
latitude  of  the  body  with  respect  to  the  plane  of  xy;  then  we  shall 

have 

x  =  r  cos  (3  cos  ^v, 

y  =  r  cos  /5  sin  w,  (60) 

g  =  r  sin ,?. 

Let  us  now  denote  by  X,  Y,  and  /,  respectively,  the  forces  which  are 
expressed  by  the  second  members  of  the  equations  (1),  and  the  first 
two  of  these  equations  give 


(Ix 


4!-!'§=/(^^-^!'>*+^- 


C  being  the  constant  of  integration.     The  equations  (50)  give 
dx  d(r  cos  13)  .  .       dw 

-77  =  COS  W  — ^-j^  —  r  cos  /?  Sm  W  -rr* 

at  at  at 


and  hence 


dy 


d  (r  cos ,?) 


dw 
dt 


=  sm  w  —      ,  +  r  cos  /3  cos  to  jj, 


dy         dx        ,      ,r,  dio 
X-J7  —  y-jT  =  r  cos"  /3  -jr- 
dt       ^  dt  dt 


tf 


VARIATION   OF   POLAR  CO-ORDINATES. 

Therefore  wc  have 

^  dw 


463 


r'  co3»  fi  yj^-  ^J(  Yx  —  Xy)  dt  +  C. 


If  wo  denote  by  Sq  the  eoniponetit  of  the  disturbing  force  in  a  direc- 
tion perpendicular  to  tlie  disturbed  radius-vector  and  parallel  with 
the  plane  of  xy,  we  shall  have 


and 
Therefore 


X  =  —  S„  sin  IV,  Y^=^Sa  cos  to, 

Yx  —  Xy  =  Sf,r  cos  /3. 
dio 


r'  cos'  /?  -~  =  r^S;  r  cos  (3  dt  +  C. 

In  the  undisturbed  orbit  we  have  /9  =  0,  and 

,  du 


"  dt 


kVpo{i-{-m); 


and  thus  the  preceding  equation  becomes 

r'  cos' ,5  — -  =  j  Sg  r  cos /?  (Zi  +  ^y^Poi^  +  '«■)• 
The  equations  (1)  also  give 


(51) 


1  _  xd\v  H-  yd'^y  +  ^dh      P(l-fm)  ^  j^«  ,    y^-^  4- zi        (52) 

If  we  denote  by  R  the  component  of  the  disturbing  force  in  the 
direction  of  the  disturbed  radius-vector,  we  have 


We  have,  also, 


R  =  X-i-Y'l  +  Zt 
r  r         r 


(53) 


xd'x  +  ydhj  +  zdh  =  d  (xdx  +  ydy  +  zdz)  —  (rfx'  +  dy^  +  dz') 
=  d  (rdr)  —  (rfr'  +  r'rfy')  =  rdh'  —  r'rfi;', 

D  denoting  the   true   anomaly   in    the   disturbed    orbit,   or,   since 

du- --=  con' i3  did" -{- d^\ 

xd'x  -f  yd'y  +  zdh  =  rd'r  —  r'  cos"  ^  dw^  —  r'rf/j\ 
Hence  the  equation  (52)  becomes 


dh- 


>rfio' 


-rcos'/5-^.---r^  + 


di5'   ,    ^'(1-1-7/0 


R. 


(54) 


464 


TIIEOUKTK'AI.   ASTUONOMY. 


170.  The  cq'.iatlons  (51)  and  (54),  in  coiinoction  with  the  last  of 
equations  (1),  com])lctely  rcproscnt  the  motion  of  a  hoavculy  body 
about  the  sun  when  u(!tc(l  upon  by  disturi)inj;  forces,  and,  when  cnni- 
ph'tely  inU'j^ratt'd,  they  will  j^ive  the  vahies  of  ?/',  /•,  and  z  for  any 
point  of  the  orbit;  but,  sineo  they  cannot  be  integrated  directly,  we 
nuist,  as  in  the  case  of  the  recitungular  co-ordinates,  find  the  equations 
whicli  give  by  integration  tlie  vahies  of  uio,  dr,  and  z,  lu  the  case 
of  the  undisturbed  orbit,  we  have 


r - 


,  (ho„ 


°    (It 


kl/p„{l  +  m), 


clt^ 


_  dw„'       Ic'd-^m)  _ 


(55) 


d( 


]-  + 


0. 


If  we  denote  by  (Iw  the  variation  of  to  arising  from  the  action  of  the 
disturbing  force,  we  have  ?«  =  «'y  +  (Jw;  and  hence  we  easily  fuid, 
from  (51), 

d8io  1        r.,  ou      li  h'     \lVpJlT~m)    .rn 

dt        r'cos'iij  \         r' cos', if  r^' 


We  have,  further, 
which  gives 

Let  us  now  put 


r'  ^  r,'  +  2r,dr  +  Sr', 


\         'o 


and  we  have 


f>n'  —  1 .S>. 

J    '1     ■•■    !1  1    C' 


1  +  25' 
The  equation  (56),  therefore,  becomes 


d'hv 
"dV 


""  r^^'7if^'> ''  ^°'^  '^  '^'  ~  ^''•^'^'' 


in  which  we  put 


ffo 


dt~  r' 


(57) 
(58) 

(59) 
(60) 


If  we  substitute  r,,  +  dr  for  r  in  equation  (54),  and  combine  the 
result  with  tlie  second  of  equations  (55),  we  get 


d'<h' 
~dt' 


R 


^oV„+rcos^3^|  +  r^'^'  +  ^'(l  +  "0(^,-J-,)i 


VARIATION   OF  POLAR   C0-0RT)n\ATE8. 


and  if  wo  put 


wo  liave 


^'-"^^V'^'^r, 


f'q"  =  l-'-±, 


r= 


l  +  2q' 


')«'"'' 


465 

(Gl) 
(62) 

(03) 


and  lience 

-"'"■■'(-+*T+'l*'i'+-(^)' 

Finally,  we  liavo,  from  the  last  of  oqiuitions  (1), 

d'z__„      kHl+on) 

^-^-— ^ 2,  (04) 

by  moans  of  whioh  the  value  of  z  may  be  found,  since,  in  the  case  of 
tlio  niidisturljcd  motion,  we  have  ::g  ~  0. 

The  values  of/'  corresponding  to  different  values  of  r/  may  be 
tabulated  with  the  argument  </,  and,  since  the  O(iuation  (02)  is  of  the 
same  form  as  (58),  the  same  table  will  give  the  value  of/"  when  r/"' 
IS  used  as  the  argument.  Table  X VII.  gives  the  values  of  lo-r/  or 
log/'  corresponding  to  values  of  r/  or  (f  from  -  0.03  to  +  0  03 
Beyond  the  limits  of  this  table  tlie  required  ciuantities  may  be  eom- 
putod  directly. 

171.  AVhen  we  consider  only  terms  of  the  first  order  with  respect 
to  the  disturbing  force,  we  have 

/V=/Y  =  ^', 

and  the  equations  become 

'2-  1.2  n      ]         \ 


dh 


df 


y       ^-'(I  +  W) 


In  determining  the  perturbations  of  a  heavenly  body,  we  first  con- 
s..lei  only  the  terras  depending  on  the  first  power  of  the  disturbino- 
toroe,  for  which  these  equations  will  be  applied.     The  value  of  dr 

30 


■tor, 


Ti r r,( )Rr:Ti( 'AL  astiu )N( )m v. 


will  1)0  obtaiiK'tl  fVuMi  the  mccoikI  cquiition  by  n"  iiKlircct  procoss,  as 
aliviidy  illiistriito(l  lor  tlio  case  of  the  variation  ol'  the  ivctaiij^nilar 
co-onlinafcs.  'i'licii  dm  will  Ix?  ohtaincd  directly  from  tlio  lir.'-it 
c'(|iiiitioii,  and,  finally,  r  indirectly  from  the  last  o(|uation.  Each  of 
tlic  intcj^rals  is  cqnal  to  zero  for  the  date  /„,  to  which  the  osculating 
elenientrt  belonj:;. 

When  tlu!  inaj^nitude  of  tlie  perturhations  is  such  that  the  terni.s 
dcpcndinif  on  the  squares  and  products  of  the  masses  must  be  con- 
sidered, the  jfcneral  ecpiations  (oO),  (03),  and  (04)  will  be  ajjplicd. 
The  values  of  the  perturbations  for  the  dates  preeedin^'  that  for 
which  the  complete  expressions  are  to  be  used,  will  at  once  indicate 
approximate  values  of  dio,  dr,  and  z;  and  with  the  values 


}•  =  ,-„  +  or, 


w  =  Wq  -\-  ihv, 


sm  /3  =  -, 


the  components  of  the  disturbin<>;  fon^e  will  be  computed.  We  compute 
also  (/  i'rom  the  first  of  eipiations  (57),  and  q"  from  the  first  of  ((U); 
then,  by  means  of  Table  XVII.,  we  derive  the  corresponding  values 
of  log/"'  and  lug/".  The  coeiHcients  of  or  in  the  expressions  lor 
fj  and  (/'  will  be  given  with  snfllicient  accuracy  by  means  of  the 
approximate  values  of  dr  and  sin ,%  and  will  not  require  any  further 
correction.     Then  we  compute  <S„r  cos/9,  and  find  the  integral 


I  S^r  cos,?  dt; 


ihhv 


and  the  complete  value  of  --.t-  will  be  given  by  (59).  The  value 
of  —7,7/  will  then  be  given  by  equation  (63).     The  term  r(  -ir  I   will 

always  be  small,  and,  unless  the  inclination  of  the  orbit  of  the  dis- 
turbed body  is  large,  it  may  generally  be  neglected.    Whenever  it  shall 

1/  dz  \'^ 
be  required,  we  may  put  it  e(pial  to  -I    ,,  I  ■    The  corrected  values 

of  the  differential  coefficients  being  introcUiced  into  the  table  of  inte- 
gration, the  exact  or  very  approximate  values  of  dio,  8r,  and  ;:  will 
be  obtained.  Should  these  I'csults,  however,  differ  much  from  tiie 
corresponding  values  already  assumed,  a  repetition  of  the  calcnlatiou 
may  become  necessary.  In  this  manner,  by  computing  each  j)laoe 
separately,  the  terms  depending  on  the  squares,  products,  and  higher 
powers  of  the  disturbing  forces  may  be  included  in  the  results.  It 
will,  however,  be  generally  possible  to  estimate  the  values  of  die,  dr, 


VAniATTOX   OF   POLAR   ro-ORDINATEH. 


467 


rind  s  for  two  or  tlirw;  iiitorviils  in  iidviuico  to  u  flcjjroe  of  approxi- 
iiiatioii  siilliciciit  fur  tli(!  connjiitiitioii  of  tlu?  forces  for  tljcso  dates. 

Ill  ord(!r  that  tlio  i|uaiitity  (o,  rcprt'scntiiifj;  tlio  interval  udoptcd  in 
tlic  calciilatiim  of  the  |icrtiii4iatioiis,  may  not  appear  in  the  iiite^ra- 
tiiiii,  we  should  intnidiK.'e  it  into  the;  e(Hiatioiis  as  in  the  ease  of  the 
variation  of  tlie  reetaiit:,iihir  co-ordinates.     Thus,  in  the  deterniina- 

1  -s 

tion  of  iiw  we  eoinpute  the  vahies  of  <n     ,— .  and  since  the  second 
'  (It 

niend)cr  of  the   e(|iiation   contains  the  intej^ral  I  S^rcotiji  (it,  if  we 

introduce  the  factor  w^  under  the  sign  of  iiitcffration,  tliis  iiitej:;ral, 
Diiiittinj:;  the  factor  o)  in  the  forinnhe  of   inte<j;ration,  will  heconie 

w I  .S'„c  cos ;9  <//,  as  rc([uired.     The  last  term  of  the  e(piation  will  l)e 
».' 

inultiplied  by  (o. 

.  d  'h' 

In  the  case  of  or,  each  term  of  the  equation  for        -  must  contain 

the  factor  w-.  If  the  second  of  ef[uations  ((Jo)  is  emi)loyed,  tlie  first 
and  third  terms  of  the  second  member  will  be  multii)lied  by  or;  but 
since  the  value  of  .S',  is  suj»posed  to  be  already  inultii)lied  by  or,  the 
SLCond  term  will  only  be  nuiltiplied  by  oj. 

The  perturbations  may  be  conveniently  determined  either  in  units 
of  the  seventh  decimal  place,  or  expressed  in  seconds  of  are  of  a 
circle  wliose  radius  is  unity.  If  they  are  to  be  expressed  in  sctconds, 
the  factor  s  =  20G2G4.8  must  be  intro(biced  so  as  to  preserve  the 
lionioy-eneitv  of  tlii'  several  terins,  and  finallv  ^/'  and  <h  nuist  be  con- 
verted  into  their  values  in  terms  of  the  unit  of  space. 

172.  It  remains  yet  to  derive  convenient  formuUe  for  the  deter- 
mination of  the  forces  H^^,  R,  and  Z.  For  this  purpose,  it  first  becomes 
lUTcssary  to  determine  the  ])osition  of  the  orbit  of  the  disturbinj^ 
planet  in  reference  to  the  fundamental  plane  adopted,  namely,  the 
plane  defined  by  the  osculating  elements  of  the  disturbed  orbit  at  the 
instant  t^y  Let  U  and  Q,'  denote  the  inclination  and  the  hjiigitude  of 
the  ascending  node  of  the  disturbing  body  with  respect  to  the  ecliptic, 
luid  let  /  denote  the  inclination  of  the  orbit  of  the  disturl)ing  body 
with  respect  to  the  fund.iuur.tal  plane.  Further,  let  A'^  denote  the 
longitude  of  its  ascendinj  nnd.  du  the  same  plane  measured  from  the 
a-cciidiiig  node  of  this  pifiae  o,i  the  ecliptic  or  from  the  point  whose 
limgitude  is  Q,^,  and  let  h"  be  he  angular  distance  between  the  as- 
ci'iKling  node  of  the  orbit  of  the  disturbing  body  on  the  ecliptic  and 
the  ascending  node  on  the  fundamental  plane  adopted.  Then,  from 
the  spherical  triangle  formed  by  the  intersection  of  the  plane  of  the 


1^  v^ 


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Photographic 

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23  WEST  MAIN  STRKT 

WIBSTH.N.Y.  14510 

(716)  •72-4503 


V 


C^ 


6^ 


168 


THEORETICAL   ASTKOXOMY. 


ecliptic,  the  fui.daniontal  piano,  ami  the  plane  of  the  orbit  of  the  dis- 
turbing body  witij  the  celestial  vault,  we  have 


sin  Usin  >, (N  +  iV")  ^  sin  A( ft'  —  R,)  sin  \  (V  +  /,), 
sin  }, I  cos  UN+  N')  ----  cos  \{Si' ~  Slo)  «»"  i  (*"  —  »«). 
cos  Usin  .J  ( iV  —  iV')  =.^  sin  ^  (  ft '  —  ftj  cos  .1  (i'  +  ig), 
cos  .J  / cos  A  ( iV  —  iV')  =  cos  .1  ( ft '  —  ft  j)  cos  ^  (t'  —  i^), 


(60) 


from  which  to  find  A',  N',  and  /. 

Let  ,y  denote  tlu;  helio<rentric  latitude  of  thi  disturbing  planet 
with  respect  to  the  fundanjental  i)]ane,  */./  its  longitude  in  this  piano 
measured  from  the  axis  of  .v,  as  in  the  case  of  w,  and  n^'  the  argu- 
ment of  the  latitude  with  res[)ect  to  this  plane.  Then,  according  to 
the  equations  (82),,  we  have 


tan  (w'  —  iV)  =  tan  «/  cos  /, 
taU/5'  =tau/sin(/  —  N). 


(67) 


If  m'  denotes  the  argument  of  the  latitude  of  the  disturbing  planet 
with  respect  to  the  ecliptic,  we  luive 


N'. 


(68) 


This  formula  will  give  the  value  of  Jt^',  and  theu  w'  and  /9'  will  1)0 
found  from  (67).     We  have,  also, 

cos  «o'  =  cos  /5'  cos  («/  —  N), 

which  will  serve  to  indicate  the  quadrant  in  which  w'  —  N  must  bo 
taken. 

The  relations  here  derived  arc  evidently  applicable  to  the  case  in 
which  the  elements  of  the  orbits  of  the  disturlxHl  and  distiu'biiig 
planets  are  referred  to  the  equator,  the  signification  of  the  quantities 
involved  being  properly  considered. 

The  co-ordinates  of  t.io  disturbing  planet  in  reference  to  the  piano 
of  the  disturbed  orbit  at  the  instant  /,,  as  the  fundamental  plane  will 
be  given  by 

af  =  r  cos  y  cos  w\ 

y'  =  r'cos,5'8ini«',  (69) 

«'  =  i'  sin  /S*. 

To  find  the  force  i?,  we  have 


r  r         r 


VARIATION  OP   POLAR  CO-ORDINATES. 


469 


and 


—  N  must  be 


Siibj<titiiting  in  those  the  values  of  x',  //',  :'  givon  by  (69),  antl  the 
corresponding  values  of  .r,  y,  z  given  by  (50),  and  putting 


we  get 


h      1        1 


(.70) 


E  =  m'k'  I  h  r'  cos  /S'  cos  <?  cos  (to'  —  iv)  ■}-  h  r'  sin  fi  sin  ,i'  —  —^  y  (71) 
The  equation 
gives 


*;rco3,9=-  Yx  —  Xy 


(SJ,  =  m'k*  h  r'  cos  /»'  sin  («;'  —  «;), 


(72) 


(73) 


from  whicli  to  find  >%.     Finally,  we  have 

Z=.„i'/t'//ir'sin/—  ^-X 

from  which  to  find  Z. 

When  we  determine  the  perturbations  only  with  respect  to  the 
first  power  of  the  disturbing  force,  the  expressions  for  li,  >%,  and  Z 
become 

R  =  wi'^-'  (  h  r'  cos  .r  cos  (ru'  —  «'.)  —  ~-A, 

Sg  ^=^  m'k*  h  r'  cos  [^  sin  {w'  —  tt',), 
Z  =  jft'P  h  r'  sin  (5'. 

To  compute  the  distance  p,  we  have 

,,» =  (x' -  x)' -4- (y  -  2/)' +  («' - «)', 

which  gives 

^»  =  /•''  ^  r'  —  2r  ?•'  cos  ,J  cos  /J*  oos  iw'  —  w)  —  2r  >•'  sin  /S  sin  ,5',  (75) 

ami,  if  we  neglect  terms  of  the  second  order,  we  have 

/,„»  =  ,•"  +  r,'  —  2ro  r'  cos  [:f  cos  (u-'  —  u\).  (76) 


(74) 


If  we  put 
we  have 


cos  Y  ^=  cos  /5  cos  (^'  cos  iw'  —  w)  +  sin  ;5  sin  ,J', 

p»  =;  r'  -f  ***  —  2rr'  cos  y 
=  r'*  sinV  -f  (»•  —  »*'  cos  j*)' ; 


(77) 


470 


THKOnETI('AI<   ASTKOXOMY. 


and  hence  wc  may  readily  find  ft  (Voni 


p  8in  II 
p  cos  n 


r  —  r'  von  y, 


(78) 


the  exact  vahic  f)f  the  angle  »,  howcN'er,  not  heing  rccinired. 
Introducing  y  into  the  expression  for  A',  it  heeonies 


by  means  of  which  U  may  be  conveniently  determined. 


(79) 


\T'\.  When  we  neglect  the  terms  dependi.ig  on  the  sqnaros  niid 
higher  powers  of  the  masses  in  the  <'ompntation  of  the  pertnrlKitions, 
the  forces  /^,  »S'y,  and  /will  be  computed  by  means  of  the  e<piatioiiii 
(74),  \t^  being  found  from  (7<3)  or  from  (78),  when  we  put 

COS  Y  --^  cos  /S'  cos  {v}'  —  «'o). 

But  when  the  terms  of  the  order  of  the  S(piare  of  the  disttu'hing 
force  are  to  be  taken  into  awount,  the  complete  ecpiations  nuist  Ih; 
used.  Thus,  we  find  />  from  (78),  «Sy  from  (72),  /from  (7.'J),  and  li 
from  (71)  or  (79).  The  values  of  f?H',  5/',  and  2,  computed  to  tlie 
point  at  which  it  becomes  necessary  to  consider  the  terms  of  tlic 
scc(»nd  onhM",  will  enable  us  at  once  to  estimate  the  values  of  tlio 
perturbations  for  two  or  three  intervals  in  advance  to  a  degree  of 
approximation  sufliicient  for  the  cidculation  of  the  forces;  and  tlio 
values  of  iif,  N„,  and  /thus  found  will  not  require  any  further  cur- 
rcetion. 

When  the  places  of  the  disturbing  planet  are  to  be  derived  from 
an  ephemeris  giving  the  heliocentric  longitudes  and  latitudes,  the 
values  of  ft'  and  /'  will  be  obtained  from  two  places  separated  by  a 
considerable  interval,  and  then  the  values  of  u'  will  be  deteriniried 
by  means  of  the  first  of  equations  (82),  or  by  means  of  (85),.  Wlicn 
the  inclination  V  \i  very  small,  it  will  be  sufficient  to  take 

«' =  r  —  ft '  4- « tan' J J' fin  2  (r  —  ft'), 

in  which  «  -  206264.8.  But  when  tlie  tables  give  directly  the  lon- 
gitude in  the  orbit,  u'  \-  ft',  by  subtracting  ft'  from  each  of  those 
longitudes  we  obtain  the  required  values  of  «'. 

It  should  1)0  observed,  also,  that  the  exact  determination  of  the 
values  of  the  forces  requires  that  the  actual  disturlMxl  values  of  /', 
%c\  and  /3'  should  be  used.     The  disturbed  radius-vector  r'  will  Iw 


VAKIATIOX   OF    l'<U-.\U   CO-OKIUXATIvS. 


471 


jijivon  iininodiiiti'ly  by  tlii"  tahlcs  ot*  tlu'  motion  of  tlit;  <listurlMii^' 
ImmIv,  Imt  llio  (Ictcrininatioii  of  the  actual  values  of  «•'  and  {i'  rc- 
qnirc's  that  wf  slioultl  use  the  actual  values  of  .V,  ,V,  an«l  /  in  the 
solution  of  the  equations  (GS)  and  ((57).  Hence  the  tlisturhed  values 
of  ft'  and  /'  should  he  used  in  the  determination  of  these  (juantities 
fur  each  date  hy  means  of  (({(j).  It  will,  however,  j;onerally  he  the 
cii-ie  tii.it  for  a  moderate  perioil  the  variation  of  ft' and  /'  may  he 
n('<;lectetl ;  and  whenever  the  variation  of  either  of  these  has  a  sensi- 
l)le  etfeet,  we  may  comptitc  new  values  of  X.  X',  and  /  fron>  time  to 
time,  l)y  means  of  which  the  true  values  may  l)e  readily  interpolated 
for  each  date.  We  may  also  determine  the  variations  of  jN',  X',  and 
/  arisinj^  from  the  variation  of  ft'  and  /',  by  means  of  diiferential 
formula?.  Thus  the  relations  will  he  siniilar  to  those  f^iven  hy  the 
equations  (71)^,  so  that  we  have 


<JiV'  = 


sin  N' 


sin  (ft'—  ft,) 
sin  N 

«in(ft'"  ftj 

.5/    =  ^in  .V  sin  /'  .)ft'  +  cos  N'  <h\ 


COS  iV '5ft' 
cos  iV '5ft' 


sin  iV' 
sin  / 

sm  2s 
sin  1 


cos/^r, 


(80) 


from  which  to  find  HX',  HX,  and  «/. 

When  the  perturbations  are  computed  oidy  in  reference  to  the  first 
power  of  the  mass,  the  cha!i<re  of  ft'  and  /'  may  Ixf  entirely  nci^- 
krted;  hut  when  the  perturbations  arc  to  be  computed  fi)r  a  lonir 
pcriiwl  of  time,  and  the  terms  dependinj^  on  the  squares  and  products 
of  the  disturbiufi  forces  are  to  he  included,  it  will  he  advisable  to 
take  into  account  the  values  of  m.V,  iiX\  and  «/,  and,  \\s'\\\\r  also  the 
value  of  h'  in  the  actual  orbit  of  the  disturbing  body,  compute  the 
actual  values  of  w'  and  ,V. 

In  the  case  of  several  disturbing;  bodies,  the  fitrccs  will  he  deter- 
mined for  each  of  these,  and  then,  instead  of  l!,  N,„  and  Z,  in  the 
fiirmuhe  for  the  differnntial  coeHicients,  27^,  -.S^,  and  -/will  be  used. 

174.  By  means  of  the  values  of  i)w,  ^r,  and  z,  the  heliocentric  or 
tlio  jjeocentric  place  of  the  disturbed  planet  m!»y  he  readily  found. 
Thus,  let  the  positive  axis  of  ,r  be  directed  t'  the  iuseendiuf^  node  of 
the  asculatii.fj  orbit  at  the  instant  /„  on  the  plunc  of  the  ecliptic; 
then,  in  the  undisturbed  orbit,  wo  shall  have 

«'„  =  n„ 


u  denoting  the  argument  of  the  latitude.     Let  a?„  y„  z,  be  the  co-or- 


172 


THKOUKTK  AL   AHTItONOMV. 


(liimtes  of  tlic  Imdy  referred  to  ji  syHtem  of  rc<'tanmilttr  co-ordinates 
ill  wliieh  the  eeliptic  in  tiie  plan*'  of  .ry,  and  in  wliieli  tiie  positive 
axis  of  X  in  directed  to  the  vernal  e(|uinox.     Then  we  hIhiU  have 

x,  =  x  cos  g^„  —  If  cos  /„  sill  fi„  -I   r  sin  /„  sin  ft„, 

t,—y  sin  t„  +  z  cos  /j, 

or,  introdncing  the  values  of  x  and  //  fjjiven  by  (50), 

X,  =  r  cos  /J  COM  w  COS)  Ji,  —  r  cos  ,'i  sin  w  cos  ig  sin  JJo  +  ^  f^hi  /„  sin  JJ ,, 

y,  =r  cos  /J  cos  h^  sin  fto  +  ''  ^^>^  i^  >*'" '"  ^"O"*  'o  ^''"'*  fto  —  ^  '■*'"  'o  *^'"*'  ft o»  "^^' 

2,  =  r  cos  ,5  sin  «» sin  /„  +  -  <'*>8  'o- 

Introdiiein^  also  the  auxiliary  constants  for  the  ecliptic  according  to 
the  equations  (94),  aiul  (9<>)i,  wo  obtain 


X,  =  r  cos  /S  sin  «  sin  (A  -\-  w)  +  «  cos  a, 
y,  ^=r  cos  ,5  sin  b  sin  (/i  +  "')  +  2  cos  6, 
z,  =  ?•  cos  /}  sin  /,  sin  »<;  -\-  z  cos  /j, 


(82J 


by  means  of  which  the  heliocentric  co-ordinates  in  reference  to  the 
ed'ptie  may  be  determined. 

If  the  place  of  the  disturbe<l  body  is  required  in  reference  to  the 
e(|uator,  denoting  the  heliocentric  co-ordinates  by  x,„  y,„  z,„  and  the 
obli(piity  of  the  ecliptic  by  s,  we  have 

Xff  =  Xf 

y„  =  y,f'oss  —  z,H'me, 
2„  =  y,  sin  £  +  2,  cos  e. 

Substituting  for  x„  y„  z,  their  values  given  by  (81),  and  introducing 
the  auxiliary  eonstant.«i  for  the  equator,  according  to  the  equations 
(99),  and  (101),,  we  get 

x„  =  r  cos /?  sin  n  sin  (A  -\-  «»)  -\-  z  cos  «, 

y„  =;  r  cos  ,J  sin  6  sin  (\B -j- «0  +  2  t-'os  ft.  (83) 

z„  =  r  cos  (J  sin  e  sin  (  C  +  w)  +  z  cos  c. 

The  combination  of  the  values  derived  from  these  equations  with  the 
corres[X>nding  values  of  the  co-ordinate»s  of  the  sun,  will  give  the 
required  geocentric  places  of  the  disturbed  body.  These  equations 
are  jipplictvble  to  the  case  of  any  fundamental  [)lane,  provided  that 
the  auxiliary  constants  a,  A,  b,  B,  ifee.  are  determined  with  respect 
to  that  plane.  In  the  numerical  application  of  the  formuhe,  tlie 
value  of  w  will  be  found  from 


VARIATION   OF   POLAR  CO-ORDINATES. 


473 


>ns  with  the 


i/„  iH^injr  tlio  arjjiiinont  of  tlio  latitude  for  the  fuiuliiint'iital  (»s<'ul!itiiiK 
I  liiiiciits,  and  t-jiro  iimst*  bo  taken  that  the  proper  alj^i'braie  >igii  i.s 
!i«si!.riie(l  to  voHO,  COS  6,  and  co.sc. 

If  the  vahies  of  z^,  Q^,  and  /„  uxod  in  the  caleuhition  of  tlie  per- 
tin'l)ationH  are  referrcnl  to  the  eeliptie  an<l  mean  etpiinox  of  the  date 
',/,  and  the  reetanfj^uhir  (to-ordinates  of  the  disturbed  Ixidy  are  re<|uir(.Hl 
ii)  relereneo  to  the  irliptie  and  mean  e<|uinox  of  the  (hite  t„",  the 
vaUic  of  10  nui«t  be  found  from 


m;  : 


''o  +  '"o  +  ^W, 


the  value  of  <o^  referred  to  the  eeliptie  of  /„'  i)ein^  reduced  to  that  of 
/,/',  by  means  of  the  first  of  c(pj!>tions  (llo),.  Then  J2,i  iU'd  /„  shouhl 
be  reduced  from  the  (H'liptie  and  mean  e<|uiuo.\  ttf  /^'  to  the  ecliptic 
and  mean  iipiinox  of  /„"  by  means  of  the  second  and  third  of  liie 
oipiatiuns  (llo),,  and,  using  the  values  thus  found  in  the  calculation 
of  the  auxiliary  constants  for  the  eeliptie,  the  e<piations  (82)  will 
give  the  ro(iuirod  values  of  the  hoIicK'eiitrie  eo-ordi nates.  If  the  co- 
onliiiatc's  referred  to  the  mejin  eipiiuox  and  eipiator  of  the  date  t^" 
arc  to  be  deternjined,  the  proper  (corrections  haviiij;  been  ap|»lied  to 
Ji„aii»'  /'u,  the  mean  obli<[uity  of  the  ecliptic  for  this  date  will  be 
employed  in  the  determination  of  the  auxiliary  constants  d,  A,  S:c. 
with  respect  to  the  equator,  and  the  equations  (83)  will  then  give 
the  required  values  of  the  eo-ordinates. 
If  we  ditferontiato  the  ecjuations  (83),  we  obtain,  by  reduction, 

(//„ 
(It 


r  cos  /?  sin  a  cos  (^  +  '**)   //  +  sec  ,5  sin  a  sin  (^1  +  «•)    -- 


-|-  (cos  a  —  tan ,}  sin  a  sin  (^1  +  "'))  -t7> 


ill 


rf?„ 
dt 


r  coa  fi  sin  h  cos  (5  -f-  w)    .    -f  sec  /?  sin  6  sin  {B  +  w)    j- 


-f  (cos  h  —  tan  ^  sin  h  sin  (B  -f-'i'))  —.-, 

(^iv  •  fir 

r  cos  /9  sin  c  cos  (C  -\- «')  -jr  -\-  sec  ,'i  sin  c  sin  (  C  -f  if)  jf 


(84) 


-|-(co8  (-•  —  tan  (5  sin  c  sin  (C-\-  «')) 


dz 

dt' 


by  means  of  which  the  component*  of  the  velocity  of  the  disturbed 
body  in  directions  parallel  to  the  co-ordinate  axes  may  l)c  determined. 


diir 


dz 


d*<h' 


d\ 


The  values  of  —^  and  -jr  will  be  obtained  from     ..7-  and    .7  by  a 


dt 


dt 


dt' 


do 


single  integration,  and  then  we  have 


474 


TIIEOKETICAL   AhTU«)X()MV. 


tf 


(80) 


dw  _  k\  ''/>„(  1  +  »i )      «/''«'  dr        kV\  +  m       .  </'^r 

trom  winch  to  iiiui    ,,  and  -rr- 
(U  (it 

17").  FiX A Mi'iiK. — 111  order  to  illustrate  the  calculation  of  the  )mt- 
turhations  of  /■,  u',  and  z,  let  us  take  tliu  data  \(\\v\\  in  Art.  l<i(i,  ami 
deterniine  these  perturbations  instead  ut'  those  oi'  the  rectangular  co- 
ordinates. 

In  the  first  place,  we  derive  from  the  tables  of  the  motion  of 
Jupiter  the  values 

^'  -_^  98°  58'  22".7,  i'  =  1°  18'  Wf), 

which  refer  to  the  eclipti*'  and  mean  equino.x  of  1860.0.  We  fnul, 
also,  from  the  data  ^ivcn  by  the  tables  the  values  of  u'  measured 
from  the  ecliptic  of  1 800.0.  Then,  by  means  of  tlu!  formulie  (()«j), 
usin^  the  values  of  J^u  and  i^  given  in  Art.  166,  we  derive 


N=  194°  0'  49".y, 


JV'  --  :}()1°  :J8'  :il".7, 


5°  0'  .w.i. 


The  value  of  uj  is  given  by  equation  (68),  and  then  lo'  and  ,i'  are 
fou!)d  from  the  wpiations  (67).     Thus  we  have 


lliTliii  Mi'Hii  Tiiiii-.  I"i;r„                  w„  =^  ii„  \i<Kr'  lo" 

18(i:U)t"f.      I'J.O,  0.21M0S4  192°   4"_M"..')  O.THl'io  14°l,S'r)4".a 

18(14  .liiii.      Ul.O,  0.'2!)4S:!7  2(17    40  52  .2  0.7:{:t(>S  17  21    44  .2 

MmitIi    1.0,  0..'«)(>(i74  22:{     8     ."1  .9  0.7:{.'Hr)  20  2.'>     .')  .2 

Ai.ril    10.0,  0.:!l(tMf!4  2:<7   .">1    38  .3  0.7:i2:{7  2:{  28   .W  .8 

May     20.0,  0.:{242'.>8  2')1    '>'!   47  .«  0.7:Ufi4  2(5  Xi  IVl  .1 

.Iimo    2!t.O,  0.:ni»74.')  2(J4  r)9  :{0  .0  0.7;Ut8(i  29  38  44  .8 

Auk.      S.O,  0.3.')(il0l  277    10  24  .(5  0.7:UK»3  .•{2  44  41  .2 

Si'i.t.     17.0,  0.3724fi9  288   28     4  .1  0.729ir)  3.-)  51    24  .(> 

Oct.      27.0,  0.388214  298   57    16  .3  0.72823  38  58   57  .5 

Doc.       (i.O,  0.402894  308  43  48  .7  0.7272(J  42  7   23  .3 

18(i.')  .Jan.     15.0,  0.41(5240  317  .')3  .39  .1  0.72(525  45  16  43  .9 


3' 

—  0°  r38".l 
0  18  9  .1 
0   34   39  .9 

0  51      7  .(1 

1  7  -Jll  .7 
1  23  43  .5 
1   .'!!»  4t!  .:! 

1  55  :\r,  :2 

2  11  7  .') 
2  2(5  20  .:) 

—  2  41    10  .() 


The  values  of  p^  may  bo  found  from  (76)  or  (78)  a.s  already  given  in 
Art.  166. 

The  forces  R,  .S',,  and  Z  may  now  be  detertnined  by  means  of  tlic 
equations  (74),  h  being  found  from  (70),  and  if  we  introduce  the 
factor  (0^  for  convenience  in  the  integration,  as  already  explained,  wc 
obtain  the  following  results: 

uj  S„r„ill 

-f-'  0".0'2S2 
—  0  .2.'i61 


Date. 
186.3  Dec.  12.0, 
1864  Jan.  21.0, 


«^  '■'''Vo 

4-l".4608       +0".M76 
-f  1  .4223       —  0  .6757 


-f  0".0009 
+  0  .0101 


N L'.MKH1CA I-    K.\ A M I'LK. 


478 


+- .... 

Dnle. 

u'/J 

..AS;,r, 

..-'/ 

(..  j  ,\r^ll 

1W4  Marc 

h   1.0, 

-1-  I".2«ilO 

-  l".4.-il2 

-{-  0".01!»0 

— '  1".:50«5() 

Apri 

10.0, 

1    .OOlH 

2  .122t; 

0  .027:5 

:5  .10:5.") 

May 

20.0, 

0  .07»tO 

2  .tJ47;J 

0  .0:547 

r>  ..^)020 

1   <»f   tlw   jHT- 

Irt.  1(!<I,  ami 
I'taiif^iilar  co- 

June 
S'|)t. 

2!).0, 

M.O, 

17.0, 

+  0  ..".l7:i 

—  0  .04.V2 
0  .;i!»44 

2  .5«W« 

:{  .i»;.")0 

;}  .14:57 

0  .040)5 
0  .0440 
0  .0470 

H  ..'5402 
11  .4:578 
14  .(lO'ii 

Oct. 

27.0, 

0  .7180 

2  .U:\U-2 

0  .04t)«J 

17  .0(140 

Dec. 

0.0, 

1  .00!»7 

2  .'htm 

0  .0432 

20  .427;^ 

e   motion  of 

IHi.")  .lun. 

1.").0, 

—  1  .2»i74 

—  2  .008 1 

-f  0  .o;}()2 

—  22  .724.-> 

).     Wt'  liiHl, 
u'  measured 
nrmuKi'  (•)♦]), 
ivc 

5°  9'  r,(j".i. 
w'  and  ,i'  are 

$' 

—  0°  l'.3S".l 
0  18  !)  .1 
C'  ;m  :!!•  .',1 

0  ol      7   .tl 

1  7  -Jil  .7 
1  2:$  4:i  ..") 
1  ;{!)  Jfi  .;! 

1  55  ;!.")  .'J 

2  11  7  .") 
2  -Jd  I'd  .:! 

—  2  41  10  .() 

Kuly  given  in 

moans  of  tlic 
introduce  the 
explained,  we 

wf .%'•-/" 
4-'  0".02H2 
—  0  .2361 


The  intej;ral  ful  >\i\,ift  is  obtained  fri>m  tlie  .suct-cssivo  values  of'w'*'N^/', 

l)V  ujeans  of  the  formula  (•V2). 

Next  we  compute  the  values  of  the  diderential  eoetlieiei  ts  hy 
means  of  the  formula;  ((Jo).  For  the  dates  WHi  Ikw  12.0  and  1«<J  t 
Jan.  21.0  we  may  fir.st  assume  or  0,  and,  Ir  a  preliminary  inte- 
gration, having  thu.s  derived  very  u])pro.\imate  values  of  ()r  for  these 

dates,  the  values  of      .      will  be  recomputed.     Then,  commencing 

anew  the  tabic;  of  integration,  we  may  at  once  derive  an  approximate 
value  of  or  for  the  d»  ■•!  March  1.0  with  which  the  last  term  of  the 

expression  for     .      may  be  compnteil.     Continuing  this  indirect  pro- 

ee-s,  as  already  illustrated  in  the  eas(!  of  the  perturbatirms  of  tlu;  re<!- 
tangular  co-ordinates,  we  obtain  the  refpiired  values  of  the  second 

differential  ooeflReient.     In  a  similar  maimer,  the  values  of    ..,  will 

1k'  obtained.    The  values  of  -.--  will  then  be  given  diret^tly  by  means 

of  the  first  of  ecpiations  (65);  and  the  final  integration  will  furnish 
the  perturbations  rc<pjired.     Thus  we  derive  the  following  results: — 


Date. 


(It 


,ptu' 


J- 


Pz 


iw 


<5r 


ill'  dt' 

18G.'?Dec.  12.0,— 0".0423-f1".4")0i)+0".0009    — 0".00    -f-0".18 +0".00 


0  .1080     1  ..'5405 

0  .71(52  4-0  .7820 

1  .6114—0  .0455 


1 8(54  Jan.  21.0, 
Mar.  1.0, 
Apr.  10.0, 
May  20.0,  2  .4705  0  .t);544 
June  2J).0,  3  .0807  1  .7333 
Aug.  8.0,  3  .2971  2  .3752 
Sept.  17.0,     3  .1080     2  .8533 


0  .0101  0  .02  0  .17  0  .00 

0  .0183  0  .40  1  .47  0  .01 

0  .0251  1  .55  3  .53  0  .04 

0  .0;i00  3  .61  5  .54  0  .09 

0  .0326  6  .42  6  .62  0  .18 

0  .0331  9  .64  5  .98  0  .29 

0  .0311  12  ,88  4-2  .98  0  .44 


Oct.  27.0,  —2  .5425  —3  .1872  4-0  .0265  —15  .73   —2  .86  4-0  .62 


47G 


Til K< UtKTRA L   AKT H()N« )M Y. 


Dntc 


</'.lr 


<nM 


I'lO 


'"  ,U  ""  ,10  '■"  .If 

18(54  Dec.    CO,  — 1".(}44.'J  -:{".4()(n»  -j-0".(H})0  — 17".Mr)  -  Il".8«  +0".8:' 
18(55  Jan.  15.0,-0  .4511-3  .53:54-1-0  .0070-18  .02-24  .21' -|  1  .05 

It  has  already  Imvii  foiiivl  that,  (luring  tho  period  iucliuhul  In  those 
results,  the  perturhatiouM  arising;  from  the  scjuares  aiwl  pr'nliiets  of 
the  distiirhiii^  forces  are  insensihle,  and  hence  the  application  of  (ho 
complete  e(piations  for  the  forces  and  for  the  ditlerential  coellicients 
is  not  re(piired.  The  e({nations  (83)  will  f^ive,  by  means  of  the 
resnlts  for  in  -  u^  4  i)ir,  r  i\,  -\-  dr,  and  z,  the  values  of  the  helio- 
centric eo-or<linates  of  the  disturbed  body,  and  the  combination  i)f 
these  with  the  co-ordinates  of  the  .sun  will  ii;ive  the  gco<!eiitric  plat  e. 
When  wo  neglect  terms  of  the  second  order,  we  have,  u(;cordin<f  to 
the  eijuatioiLS  (84), 


dx,.  =  X.  cot  (A  -f-  «')  'hv  -f-  --  tir  -f  2  co8  a, 
8y;,  =  y,  cot  ( 2?  -\-  w)  "U'  -f-  -  <fr  -f  z  ooa  b, 
dz„  =  z„  cot  (C  -f-  w)  »5h»  -\-  —  dr  +  2  cos  c, 


(m) 


the  heliocentric!  co-ordinates  .r„,  y^,  z„  being  referred  to  the  same  fiiii- 
dainental  plane  as  the  auxiliary  constants,  a,  h,  A,  Ac.  Thus,  in  tlio 
case  of  Eut'ifiKmie,  to  find  the  perturbations  of  the  rectanfjular  co-or- 
dinates, referred  to  the  ecliptic  and  mean  equinox  of  18()0.0,  from 
18G4  Jan.  1.0  to  18G5  Jan.  15.0,  we  have 

A  =-.  29(5°  .34'  37''.5,  B  =  '2m°  4.1'  34".4,  C-^  0, 

log  coH  a  --=  8..')573.54„  log  cos  6  =  8.85074(3,  log  cos  c  —  log  cos  („  =--  9.998';90, 


log  Jo  :^  0.399807, 


log  ija  ^  9.838709, 


log  ia  =  9.148170, 


w  ==  «'o  +  <'w  -  317°  53'  20".2, 
and  hence,  by  means  of  (86),  we  derive 

dx,  =  -f  36".559,        dy,  =  -j-  41".083,        Sz,  = 


0".588. 


If  we  express  these  in  parts  of  the  unit  of  space,  and  in  units  of  the 
seventh  decimal  place,  we  obtain 


dx,  =  -f  1772.4,        Sy,  =  -f  1991.8,        dz. 


28.5, 


agreeing  with  the  results  already  obtained  by  the  method  of  the  va- 
riation of  rectangular  co-ordinates,  namely, 

Sx,  =  +  1772.6,        8y,  =  +  1992.3,        8z,  =  —  28.2. 


(IIANOE  OF  THK  OMcULATINO    KI.KMKNTH. 


477 


170.  \W  iix\u\r  (lie  roinplftt'  funnula',  llic  |M'rl'nOmtlonM  (»f  r,  «», 
mill  :  iDiiy  Im-  roiiipiitcd  witli  rc>i»('ct  t<»  all  powers  of  tlif  tlistiirl)iii>; 
forcr,  un*l  fur  u  loii^  scries  of  years,  iisiii\r  eoiistantly  the  same  I'lm- 
(laiiieiital  nseiilatiti^  eleinciits.  iSiit  even  when  tliese  elenient.H  are  ho 
neeiimte  as  not  to  re<|uirt'  eorreetion,  on  tUT'onnt  of  the  efloet  of*  the 
large  pertnrhations  of  lonj;  period  npon  the  values  of  oir  anil  nr,  the 
iiuniitrieal  values  of  the  ]ierturl)ati<)ns  will  at  length  Ih;  sueh  that  u 
•  liaiiige  of  the  oseulating  elenients  heeonies  desirahle,  so  that  the 
integration  may  again  commence  with  the  value  xcro  for  the  variation 
of  Oifeh  of  the  co-ordinates.  This  chang(>  from  ono  system  of  ele- 
ments to  another  system  may  be  readily  cU'ected  when  the  values  of 
the  perturbations  are  known.  Thus,  having  found  tlii'  distuvbei! 
values  of  /•,  w,  and  s,  we  have 


p  being  the  semi-parameter  of  the  instantaneous  orbit  of  the  disturbed 
hodv.     In  the  undisturbed  orbit  we  have 


170 


_</«•,_  H>.,'1  +»h) 


dt 


'0 


p^r*    rfc' 


and  hence  we  derive 

Substituting  for    ..  the  value  above  given,  there  results 


(87) 


1  a 

by  means  of  which  jj  may  be  determined.     To  find  -4ti  wc  have 


dt 


We  have,  also. 


rf;J 1        dz       tan;J    dr 

dt       rcos,?    dt  r       dt 


(88) 


lit  ~  ~17p 


k\/l-\- 


VI 


iiff  •  rv  v     a.  ~i~  lit  ■  ,      itOr 

—  e  sin  V  =        ^—  -  €o  sin  v^  -f-  -.-, 


Vp, 


and  if  we  put 


(89) 


478  TIIKOUKTKAI.   AsriloNOMY. 

tliix  equation  hoconicM 

c  sin  «  —  c„  sin  I', -f- »/■„  Hin  I'o  +  r- 
Wo  liuvo,  flirt lior, 


(90) 


P      1 


und,  putting 
we  obtain 


J'o     I' 


--  1  +  <J, 


(91) 


I  COS  y  =  fo  *'*>«  *'o  ~f" '''    • 


This  c<iuation,  combined  with  (90),  j^jives 

e  sin  (v  —  v^)  —  ar„  sin  )•„  cus  j'o  +  y  cos  v„  — '  "  ,J  sin  r^, 


i'o 


(92) 


e  cos  fv  —  vj  =  Cfl  +  a«j  sin't'o  -}-  r  «»»  'V  +  -.-  /'  cos  t',„ 


by  means  of  which  the  vahics  of  *■  and  r  may  be  found,  those  of  the 
auxiliaries  a,  [t,  j',  being  found  from  (89)  and  (91).     Then  we  have 


e  =  sui  <f>. 


kVl+m 


03 


a  =  ;j  sec'  <f, 
tan  \  E  —-  tan  (4o°  —  \ip)  tan  Jv, 
il/=£  — esin£, 


by  means  of  which  tp,  n,  ft,  and  M  may  be  determinetl.     In  the  r.ise 
of  orbits  of  great  eccentricity,  we  find  the  perihelion  dit»tance  from 

P 

'       1  -f-  e 

and  the  time  of  j)orilielion  passage  Avill  be  derived  from  c  and  i'  by 
means  of  Table  IX.  or  Table  X. 

It  remains  yet  to  determine  the  values  of  Q,  i,  and  <o  or  r.  Let 
^u  <lenote  the  longitude  of  the  ascending  node  of  the  instantaneous 
orbit  on  the  plane  of  the  osculating  orbit,  defined  by  ft^  ami  /„,  mea- 
sured from  the  origin  of  w,  and  let  j^q  denote  its  inclination  to  tiiis 
plane.     Then  we  iiave 


tan  ijo  sin  {w  —  0^        =  tan  y9, 
tan  7j^  cos  (w  —  0 J  -^  =  sec'  /?  -^, 


(93) 


and  hence 


CIIANUE  OF   TIIK  «»(  UI.ATINtl    KLKMKNTS. 


479 


tonCij;  —  0„)  —  ^  Hi  II 2,  J 


g.'¥Tf 


iLL 
til 


(04) 


l»y  nicnns  of  which  0„  may  Ik-  iiiinul.  The  <|U!iili'iiiit  in  which  (i„  i.i 
situated  is  ih'tcriiiiiicd  hy  the  «-oii(litioi)  that  siiiif/'  //„)  aiitl  taii^i 
must  have  the  same  si^^n.  The  vahie  itf  r^^^  will  be  Inuiul  from  the 
lir>l  or  the  jseectiid  of  ei|iiatioiiH  (\i'.\). 

If  we  denote  l»y  ^  the  ar^nmcnt  of  the  latitntle  of  the  disturlM-d 
iMKJy  with  respect  to  the  adopted  fun<lameiital  plane,  we  have 


tan  C  = 


Uin  (n<  —  0,)^ 


(05) 


and  the  angle  ^  must  he  taken  in  the  wiinc  quadrant  rs  io  —  d„. 
Then,  from  the  spherical  triaiijfle  formed  hy  tht;  iii(  rseeti'ii  of  the 
planes  of  the  ecliptic  and  instantaneons  orhit  of  tlu  «iistMrl<ed  'inly, 
aiul  the  fun-^iMiutal  plane,  with  the  eelestial  vnnlt,  we  (haive 


co8it8in(^(it  — :)+  UJi 
cosi  teo»(.J(«  — 0  +  \iQ, 
m\\  / sin  (! (It  — C)  —  ^(R 

mi  \  i coH {!  (it  —  J)  —  n  Ji 


-  Jio')  '"'-  CO«  •i^'n  l'««  J  *  «o  +  %). 


(06) 


These  ocpiations  will  furnish  the  values  of  /,  u  —  ^,  and  ft  ~  ft,,,  and 
hence,  since  ^  and  ft,,  are  j;iven,  those  of  ft  and  it.  The  value  of  v 
having  been  already  found,  we  have,  finally, 

at  =  u  —  V, 

ff  =:|t  —  V+   ft, 

ami  the  elements  arc  eomplet<'ly  determined.  These  elements  will 
he  referred  to  the  celipti<;  and  mean  e<|uinox  to  which  ft„  and  /„  are 
referred,  and  they  may  be  reduced  to  the  equinox  and  ecliptic;  of  any 
otiier  date  by  means  of  the  formulte  which  hav(!  alrcaily  l)een  given. 
The  elements  of  the  instantaneous  orbit  of  the  disturl)ed  body  may 
also  be  determined  by  first  computing  the  values  of  ./•„,  _j/,„  z,„  in 
reference  to  the  fundamental  plane  to  which  ft  and  /  are  to  be  re- 
ferred, by  means  of  the  equations  (83),  and  also  those  of  -,yt    ^,'j,  — -' 

l)y  means  of  (85)  and  (84),  and  then  determining  the  elements  from 
the  co-ordinates  and  velocities,  as  already  explained. 
It  should  be  observed  that  when  the  factor  w',  or  the  scjuare  of  the 


4«<) 


TIIKOUKTICA r.    AKTKONOM V. 


{ 


iuloptcd  iiiu  rviil,  is  introduced  into  tlio  (>xjin'srtion«  for  the  forces  and 
ditU'rcntial  cot'lHcients,  the  first  integrals  will  l)o 


Ut' 


di' 


and  that  when  these  )|nuntitles  are  expressed  in  seconds  of  arc,  (hoy 
ninst  he  converted  i!it()  their  valnes  in  parts  of  the  nnit  of  spaco 
whenever  they  are  to  he  eonihined  with  (piantities  which  an?  not  cx- 
prcsstnl  ill  seconds.  In  other  words,  the  homogeneity  of  the?  several 
terms  must  be  cairefully  attended  to  in  the  actual  applic^ation  of  the 
forniuhe. 

When  the  elements  which  correspond  to  given  vahies  of  the  per- 
turhations  have  been  determined,  if  we  compute  the  heliocentric 
longitU(h>  and  latitude  of  the  body  for  the;  instant  to  which  the  ele- 
ments belong,  the  results  should  agree  with  those  obtained  by  com- 
puting th(!  heliocentric  place  from  the  fundamcntul  osculating  ele- 
ments and  adding  the  perturbations. 

177.  The  (!omputation  of  the  indirect  terms  when  the  perturha- 
tions  of  the  co-ordinates  r,  w,  and  s  are  determined,  is  cHected  with 
greater  fiicility  than  in  the  rase  of  the  rectangular  co-ordinates, 
although  {\w  liiial  results  are  not  so  convenient  for  tin;  calculation  of 
an  ephemeris  for  the  conip.  *  on  of  observations.  This  indinu't  cal- 
culation, which,  when  the  pertin-bations  of  any  system  of  t.hre(>  co- 
ordinates are  to  be  computed,  cannot  in  any  case  be  avoidcsd  witlioiit 
impairing  the  accuracy  of  the  results,  may  be  further  simplified  l)y 
determining,  in  a  pecruliar  form,  the  perturbations  of  {\w  niciui 
anomaly,  the  radius- vector,  and  the  (M)-ordinate  ;  perpendicular  to  the 
fundamental  plane  adopted. 

Let  the  motion  of  \\w  disturbed  body  be,  at  each  instant,  referrih 
to  the  plane  of  its  instantaneous  orbit;  then  we  shall  have  /9  -  0, 
and  the  etpiations  (51)  and  (51)  become 


r''^'"  = 


■Xnrdl-ykVp^a+m), 

k'a  -\-  VI) 


dt 

d^r         div* 


(fl7) 


dt 


dt* 


--=R, 


in  which  R  denotes  the  component  of  the  disturbing  force  in  flic 
direction  of  the  disturbed  radius-vector,  and  »S'  the  (!omponent  in  the 
plane  of  the  disturlK'd  orbit  and  perpendicular  to  the  disturbed  mditis- 
veetor,  being  positive  iu  the  direction  of  tlie  motion.     The  efl'ecl  of 


VAUIATION  OF    I'OLAK  CO-OUDINATKS. 


481 


ic  forces  and 


the  ('fimpnnonts  II  and  »S'  is  to  vary  the  form  of  tlio  orbit  and  the 
iiM^nhir  distance  of  the  periiiclion  ironi  the  ncxle.  If  we  denote  by 
/  the  (H>ni|)onent  of  the  distiii'i)in}{  ioree  |)(>i'pendi<ndar  to  the  phme 
of  the  instantaneous  orbit,  the  elfeet  of  this  will  be  to  ehange  the 
po!<ition  of  the  phme  of  the  orbit,  and  henee  to  vary  the  elements 
wliieli  depend  on  the  position  of  this  |)lane.  ^ 

Jx't  us  take  a  fixed  line  in  the  plane  of  the  instantaneous  orbit, 
and  suppose  it  to  be  directed  from  the  centre  of  the  sun  to  a  point 
whose  angular  distjine(;  back  from  the  pliu'e  of  tin;  ascending  no^le  is 
(tj  and  let  the  vahn;  of  a  be  s()  taken  that,  so  long  :is  the  position  of 
the  plane  of  the  orbit  is  unchanged,  we  shall  have 

<T^-  9,. 

The  line  thus  taken  in  the  phuu;  of  the  orbit  may  be  rcgard(Ml  as 
fixed  during  all  changes  in  the  position  of  this  plane.  Let  ;f  denote 
the  angle  between  this  fixed  line  and  the  semi-transverse  axis;  then 

will 

;^  ^  w  +  <T,  (98) 

and  when  the  position  of  the  plan(!  of  the  orbit  is  unchanged,  we  have 

But  if,  on  account  of  the  action  of  the  component  Z,  the  position  of 
the  plane  of  the  orbit  is  changed,  we  have,  according  to  the  equations 
(72)2,  the  relations 


rf(T  =  cost'fZjJ, 

dm  =  dx  —  cos  idQ,, 

dn  —dx-\-{l—  cos  0  d^^dx  +  'i  sin'  Xid^. 


We  have,  further, 


n 


v-\-x~"y 


(99) 


(100) 


V  being  the  true  anomaly  in  tin;  instantaneous  orbit. 

The  two  eomjwnent.s  of  the  disturbing  force  which  act  in  the  plane 
of  the  disturlx /!  orbit  will  only  vary  ;f  and  the  elements  which  deter- 
mine the  dimensions  of  the  conies  section.  We  have,  therefore,  in  the 
cast  of  the  ost^B^iuting  elements,  for  the  instant  /„, 

Xa  ==  %  4-  Sio  =  '^0- 

Let  us  now  suppose  X  to  denote  the  true  longitude  in  the  orbit,  so 
that  we  have 

31 


482 


THEORETICAL  ASTRONOMY. 


or 


x  =  v  +  x  —  (<'—Sl); 


(101) 


then,  since  y  is  equal  to  t:  when  the  position  of  tlie  plane  of  the  orbit 
is  unohanged,  it  follows  that  a  —  SI  represents  the  variation  of  the 
true  longitude  in  the  orbit  arising  i'rom  the  action  of  the  component 
y^of  the  disturbing  force.  The  elements  may  refer  to  the  ecliptic  or 
the  equator,  or  to  any  other  fundamental  plane  which  may  Ijc  adopted. 

178.  For  the  instant  t  we  have,  in  the  case  of  the  disturbed  motion, 
the  following  relations : — 


r  cos  V  ==  a  cos  E  —  ae, 
r  sin  v  =  aVl- 


'  sin  E, 


(102) 


-U 


x-i'^-Sll 

Lot  us  first  consider  only  the  perturbations  arising  from  the  action  of 
the  two  components  of  the  disturbing  force  in  the  plane  of  the  dis- 
turbed orbit,  and  let  us  put 

^,^v-\-x-  (103) 

Further,  let  3iJ,  -f  A'oC^  ~  Q  +  ^^^  ^^  the  mean  anomaly  which,  by 
means  of  a  system  of  eq  ations  identical  in  form  with  the  preceding, 
but  in  which  the  values  of  a^,  e^,  ;f„  are  used  instead  of  the  instanta- 
neous values  a,  e,  and  jf,  gives  the  same  longitude  ^„  so  that  we  have 

E,  -  e,  sin  E,  =  M,^  ix,  {i  -  Q  +  8M, 
r,  cos  V,  =  a„  cos  E, 


%<",,' 


r,  sm  V, 

^>  =  V,+Xo 


Co'  sin  E, 


(104) 


If,  therefore,  we  determine  the  value  of  dMso  as  to  satisfy  the  con- 
dition that  ?.,=■- V  -{-  y,  the  disturbed  value  of  the  true  longitude  in 
the  orbit,  neglecting  the  effect  of  the  component  Z  of  the  disturbing 
force,  will  l)c  known.  The  value  of  r,  will  generally  differ  from  that 
of  the  disturbed  radius-vector  r,  and  hence  it  becomes  necessary  to 
introduce  another  variable  in  order  to  consider  completely  the  effect 
of  the  components  R  and  S.    Thus,  we  may  put 

r  =  r,(l  +  v),  (105) 

and  V  will  always  be  a  very  small  quantity.  When  dM  and  v  have 
been  found,  the  effect  of  the  disturbing  force  perpendicular  to  the 
plane  of  the  instanianeous  orbit  may  be  considered,  and  thus  the 
complete  perturbations  will  be  obtained. 


VARIATJON  OP  CO-ORDINATES. 


483 


In  the  equations  (97),  J/*^  ,-  expresses  the  areal  velocity  in  the  in- 
stantaneous orbit,  and  it  is  evident  that,  since  the  true  anomaly  is  not 
affected  by  the  force  Z  perpendicular  to  the  plane  of  the  actual  orbit, 

\i^  ~  must  also  represent  this  areal  velocity,  and  hence  the  equations 
become 


r*^  =fSrdt  +  kl/p,{l  +  m), 


d?r 


-K^)'+^- 


(1  +  m) 


(106) 


=  R. 


179.  If  we  differentiate  each  of  the  equations  (104),  we  get 

,.  „.dE,  .  dm 

(l-e,cosi;,)-^-=/x.  +  -^. 


dt 
dr, 


dv, 
dt 


dE, 


cos  V,  -—■  —  r,  sin  v,  -  jf  =  —  a^  sin  E,  -^ , 


(107) 


«/■,    ,  dv,  / _,  dE, 

sm  v,-T-  -\-r,  cos  V,  -^  =  a^V  1—eo  cos  L,  —r-, 

d>., dv, 

'dt    '"dt' 

From  the  second  and  the  third  of  these  equations  we  easily  derive 


dr, 
''dt 


r,  ~,r  =  (oo^l  —  V  **'  8^"  ^1  ^os  E,  —  a^r,  cos  v,  sin  E,)  -v-. 


(IE, 
dt 


dE, 


Substituting  in  this  the  values  of  r,  sin  v„  r,  cos  v„  and  -i-'.  and  re- 
ducing, we  got 

dr,         .,     '    tA       ,  dUfK 
r,-^  =  ao'^„sm£,^/.„  +  -^|, 

or 

dr,       kVT+^i       .      I,   ,    1    dS3r\  ..... 


dr, 


From  the  same  equations,  eliminating  — ,  we  get 


dt 


'   dt 
which  reduces  to 


dE, 


r,*  --f^  —  (Ool/l  —  ej'  r,  cos  v,  cos  E,  +  a^r,  sin  v,  sin  E,)  -~, 


.■1=,/,-;a+^)(i  +  i.«f),  (109) 


484 


THEORETICAL  ASTBONOMV. 


or 


Combining  this  with  the  first  of  equations  (106),  we  get 

=  I'o  (  7T-T-T,  -  1  )  +  }l-4'Si  •       ,--^— =^   fSrdt,     (110) 


mi 

dt 


from  which  dM  may  bo  found  as  soon  as  v  is  known. 
The  equation  (105)  gives 

d*r       .^    .     ..d^r,   ,   _rfr,     dv  d'v 

rf^^^^  +  ^^W  +  ^dT'W  +  '-'.F 


(111) 


dv, 

.      1*^0 


Differentiating    equation    (108)  and  substituting  for  -^  its  value 
already  found,  we  obtain 


dt^~ 


0  6,0031',  /ill    d<^M\ '      kV^l  4-  tn  f  0  sin  v,     d''J3/ 

7       ^  ■^";i;;"dr/+      ;i;;^;^        "dt^' 


and  the  last  of  the  preceding  equations  becomes 

''df  + 

+ 


dh' rf'v    I  ^"(1 +  m)  fjcost', 

df~''' 


'dt'    ■  r/ 

kVl  +  m 


<'+'>(>+i-ifr 


=- —  Cg  sm 


rfv    ,    2     rfv    d53I\ 


The  equation  (110)  gives 


/L±_!:  ^¥ x.9^M-  —  --\ 
"'l  /^o    '  «^f'   ^    rf<  "^/x/  rf<  ■  dt  J' 


1     rP<5J»f  2 


dv  2 


rfw 


Mo     rft'    ^  (1  +  ")'    rf<    '  (1  +  ")'    dt     k\/p,{l  +  jh) 

1  >Sr 


/ 


^rd« 


which  is  easily  reduced  to 


(i-i-")'  k\/p,(i+my 


and  hence  we  derive 


rfV_    d'v  ,  Jt'(l  +  m 


'  *"'  jrt 


df  ■  r, 

The  equation  (109)  gives 


O^^c^,        /     1 ,  dmy    e^ 


VARIATION  OF  CO-ORDINATES. 


485 


and,  siucc 


this  becomes 


''[dt)-~-r;^ V'^]::    dt   }' 


r 


^  =  1  +  <J,  C08  V„ 


. /  dv, \^_  ^'(1  +  «0 n  4-  vW  1  -J-  ^   '^'^^^^V 


+ 


/io     dt 


(113) 


r,' 


1^ 


~o'    dt'l 


Combining  equations  (112)  and  (113)  with  the  second  of  equations 
(lOG),  we  get 

''     1  + " p  ,  fc'(i-^m)(i  +  ^y I..1  dmy 
f^==-T-^  + 7— — V^^^-^dfj 

Co  sin  V,  ^      ^*(1  +  m)  (1  +  v) 


dt 


(114) 


Po 


'S- 


r» 


From  (110)  we  derive 


and  the  preceding  equation  becomes 

1 


di 


111      ?L   I   o^'(l  +  m) 


Vi>,(l  +  m)»/  Po 


^»(l  +  «0         P(Mim)  / 


(115) 


which  is  the  complete  expression  for  the  determination  of  v. 

It  remains  now  to  consider  the  effect  of  the  component  of  the 
disturbing  force  wliich  is  perpendicular  to  the  plane  of  the  disturbed 
orbit.  Let  x„  y„  z,  denote  the  co-ordinates  of  the  body  referred  to 
the  fundamental  plane  to  which  the  elements  belong,  and  x,  y  the 
co-ordinates  in  the  plane  of  the  instantaneous  orbit.  Further,  let  a 
denote  the  cosine  of  the  angle  which  the  axis  of  x  makes  with  that 
of  x„  and  /9  the  cosine  of  the  angle  which  the  axis  of  y  makes  with 
that  of  y„  and  we  shall  have 

«,  =  «x  +  /9t/.  (116) 

If  the  position  of  the  plane  of  the  orbit  remained  unchanged,  these 


486 


THEORETICAL   ASTRONOMY. 


cosines  a  and  ^  would  be  constant ;  but  on  account  of  the  action  of 
the  force  perpendicular  to  the  plane  of  the  orbit,  these  quantities  arc 
functions  of  the  time.  Now,  the  co-ordinate  z,  is  subject  to  two  dis- 
tinct variations:  if  the  elements  remain  constant,  it  varies  witli  the 
time;  and,  in  the  case  of  the  disturbed  orbit,  it  is  also  subject  to  a 
variation  arising  from  the  change  of  the  elements  themselves.  We 
shall,  therefore,  have 

di~\dt  1^  \_dtS 
in  which  I  -.-  I  expresses  the  velocity  resulting  from  the  constant 

elements,  and        y-      that  part  of  the  actual  velocity  which  is  due 

to  the  change  of  the  elements  by  the  action  of  the  disturbing  force. 

But  during  the  element  of  time  dt  the  elements  may  be  regarded  as 

dz     . 
constant,  and  hence  the  velocity  -jr  in  a  direction  parallel  to  the 

axis  of  z,  may  be  regarded  as  constant  during  the  same  time,  and  as 
receiving  an  increment  only  at  the  end  of  this  instant.  Hence  we 
shall  have 


dt~\dt)  \_dtj~ 


Differentiating  equation  (116),  regarding  a  and  /9  as  constant,  we 

get 

ldz,\      dz,  dx  dy  , 

\dtl'-Ht=''di  +  ^lii'  ^^^'^ 

and  differentiating  the  same  equation,  regarding  x  and  y  as  constant, 
we  get 

Differentiating  equation  (117),  regarding  all  the  quantities  involved 
as  variable,  the  result  is 


d\ 


Now,  we  have 


da     dx       di3    dy         ^**   i    s  ^V 
'dt"  ~di '^  ~dt  "dt  '^"dF  "^ '  dF' 


Z,  =  aX  +  13  Y-i-Z  COS  i, 


(119) 


(120) 


in  which  Z,  denotes  the  component  of  the  disturbing  force  parallel 
to  the  axis  of  z„  and  i  the  inclination  of  the  instantaneous  orbit  to 


VAniATION   OK  CO-ORDINATES. 


487 


the  constant 


the  fdiulatnontul  phmc.    Substituting  for  A' and  y  their  vahics  given 
by  the  e(juations  (1),  and  reducing  by  means  of  (116),  we  obtain 

or 

Comi>aring  this  with  (119),  tliere  results 

da    dx    ,    (/,?     (hi       ., 

-dt'       dt■^-dt'-dt='^''''^ 


lSl.  The  equation  (120)  gives 
kHl-\-  Ml) 


lit:' 


z,  -{■  Z  cos  i  +  ^X  +  /3  Y. 


(121) 


(122) 


The  component  of  the  disturbing  force  perpendicular  to  the  plane  of 
the  disturbed  orbit  does  not  alfijct  the  radius-vector  r ;  and  hence, 
when  we  neglect  the  effect  of  this  com])onent,  and  consider  only  the 
components  R  and  *S'  which  act  in  the  phu.c  of  the  orbit,  we  have 


dt' 


Jl-'(1  +  m) 


h-\-'*oX-\-,%Y, 


(123) 


in  which  z^  denotes  the  value  of  s,  obtained  when  wc  put  Z~-0. 
Lot  us  now  denote  by  oz,  that  part  of  the  change  in  the  value  of  z, 
which  arises  from  the  action  of  the  force  perpendicular  to  the  plane 
of  the  disturbed  orbit,  so  that  we  shall  have 

Substituting  these  in  equation  (122)  and  then  subtracting  equation 
(123)  from  the  I'csult,  we  get 


dt* 


k'il  +m) 


Sz,  +  Zcos  i  +  XSa.  -f-  r<J/9. 


(124) 


The  equations  (116)  and  (117)  give 

Sz,  =  xSa,  -j-  ydi3. 
If  we  eliminate  8^3  between  these  equations,  there  results 


dSz,       dx  d\i 

-dV  ==-dt^''+  dt  "'' 


'-\^-dt-y-di) 


dy 
dt 


Sz,—y 


doz, 

~di' 


488 


THEORETICAL   ASTRONOMY. 


/ 


and  since  the  factor  of  3a  in  this  equation  is  double  the  areal  velocity 
in  the  disturbed  orbit,  wc  have 

P^iiiiiinating  Ja  from  the  same  equations,  wc  obtain,  in  a  similar 
mannor. 

Substituting  these  values  in  equation  (124),  it  becomes 
P(l  +w) 


d'<h, 
dt* 


8z,  -\-  Z  cos  I 


+ 


kVp(\ 


hw,{{-'Ii-^->'+^^--^y^'^\ 


(127) 


If  we  introduce  the  components  R  and  S  of  the  disturbing  force,  we 
have 


and  hence 


r  r 


r  r 


dr 


x^l-y^  =  ^,VF(l+^^-S^. 


dt 
Yx     —  Xy 


■  Sr. 


Therefore  the  equation  (127)  becomes 
d*Sz,  ife'd  +  m), 


df 


-Sz,  -\-  Zcosi 


IB 8 dr  \  ^j^     

We  have,  further, 


Sr dSz, 

(1  +  m)  '  dt  ' 


dr       ..    .    .dr,    ,       dv 


(128) 


which,  by  means  of  the  equations  (108)  and  (109),  gives 
dr Co  sin  t),      ,  dv,  dv  kVpjl  +  "0 


dt  ~j9o(l  +^)'^  dt  '^'^'  dt 


i>o(r+^)~'°^^"''  +  ^'-^ 


Substituting  this  value  in  the  equation  (128),  we  obtain 


VARIATION   OF  CO-ORDINATES. 


489 


»0  .      I    «       •    ,   /-R      Co  sin r,  „\     ffz, 

—  oz,  4-  Z  COS  I  4- \ ° »S  I ,  -,^— 

\r,  po        / 1  +  " 

_5r /  (l^Zj^ oz,_     <h  \ 


+ 


(129) 


wliicli  is  the  complete  expression  for  the  determination  of  dz,, 

182.  The  equations  (110),  (115),  nml  (129)  determine  the  complete 
perturbations  of  the  disturbed  body.  The  vtilue  of  ^  must  first  l)c 
obtained  by  an  indirect  process  from  the  equation  (115),  and  then  d}[ 
is  given  directly  by  means  of  (110).  The  value  of  dz  will  also  bo 
determined  by  an  indirect  process  by  means  of  (129). 

In  order  to  obtain  the  expressions  for  the  forces  li,  S,  and  /,  let  lo 
denote  the  longitude  of  the  disturbed  body  measured  in  the  plane  of 
the  instantaneous  orbit  from  its  ascending  node  on  tlu>  I'undamental 
plane  to  which  SI  and  i  are  referred,  it  being  the  argument  of  the 
latitude  in  the  case  of  the  disturbed  motion.  lAit  w'  denote  the  lon- 
gitude of  the  disturbing  body  measured  from  the  same  origin  and  in 
the  plane  of  the  orbit  of  the  disturbed  body,  and  let  ^i'  denote  its 
latitude  in  reference  to  this  plane.  Finally,  let  N,  N',  I,  and  u^' 
have  the  same  signification  in  reference  to  the  plane  of  the  instanta- 
neous orbit  that  they  have  in  reference  to  the  plane  of  the  undisturbed 
orbit  in  the  case  of  the  equations  (6G).     Then  we  shall  have 


sin  Usin  •  (JV+  N')  =  sin  J  (ft'  —  ft)  sin  ^(i'  +  i), 
sin  Ucos  A  (N+  N')  =  cos  ^  (ft'  —  ft )  sin  J  (i'  —  i), 

cos5/sini(^— iV')==sini(ft'— Q)t'osi(i'+  0, 
cos  Ucos  ^  (iV—  N')  =  cos  A  (S^'  —  ^)  cos  \  (t'  —  i), 

from  which  to  determine  N,  N',  and  /.     Wo  have,  also, 

n^  =  u'-N', 
tan  {w'  —  N)  =  tan  »,'  cos  /, 
tan  t^  =  tan  /sin  (to'  —  N), 


(130) 


(131) 


from  which  to  find  w'  and  ,9',  w'  being  the  argument  of  the  latl-ade 
of  the  disturbing  bo<ly  in  reference  to  the  plane  to  which  ft  and  i 
are  referred. 

Since,  when  the  motion  of  the  disturbed  body  is  referred  to  the 
plane  of  its  instantaneous  orbit,  /5  =  0,  the  equations  (71),  (72),  and 
(73)  become 

Jt  z=z  m'k^  I  h  r'  cos  ,3'  cos (v/  —  w) A, 

S  =  m'k^h  r'  cos  /J'  sin  (t</  —  w), 
Z-— w'ifc'/i.r'sin,?', 


(132) 


490 


THEOUKTICAL    ASTRONOMY. 


by  incaiiy  <»f  which  the  rwiuircd  nmipoiicnts  of  the  di.sturbiiig  force 
limy  be  Ibiiiid,  the  value  of  U  being  given  by 


»=l-i 


,.'»• 


To  i'md  (I,  we  have 

fi?  — -  7-''  -j-  r*  —  2/t'  cos  ii"  cos  do'  —  w), 
cos  y  =  cos  /i*  cos  (w'  —  if), 


or,  putting 
the  equations 


p  »\n  n  =^  r  .sni  y, 

ft  cos  n=^r  —  r'  cos  y. 


(133) 


(134) 


The  vahios  of  r'  und  u'  for  the  nctual  places  of  the  disturbint,' 
body  will  l)c  <;ivon  by  the  tables  of  its  motion,  and  the  actual  values 
of  S^'  and  /'  will  also  be  obtained  by  means  of  the  tables.  The  de- 
termination of  the  actual  values  of  /•  and  w  requires  that  the  pertur- 
bations shall  be  known.  Thus,  when  ojf  and  v  have  been  ibuiid, 
we  eomputo,  by  means  of  the  mean  anomaly  J/„  +  /io(<  —  t^  -\-  d.M 
and  the  elements  o„,  c„,  the  values  of  v,  und  r,.  Then,  since 
^  _|_  j^  -^  J,  _l_  j.^^  ^y^,  iijive,  according  to  (100), 


Wc  have,  also, 


W  =  V,  4-  r.^  —  ff. 


(135) 


In  the  case  of  the  fundamental  osculating  elements,  we  Iiave 

<''o=  Slot 

which  may  be  used  as  an  approximate  value  of  o;  but  the  complete 
determination  of  to  requires  that  <t  =  J^o  +  (5<t  shall  also  be  deter- 
mined. The  exact  determination  of  the  forces  also  requires  that  the 
actual  values  of  SI  and  l  as  well  as  those  of  SI'  and  i',  shall  be  u^ed 
in  the  determination  of  iV,  N',  and  I  for  each  instant.  When  these 
have  been  found,  it  will  be  sufficient  to  compute  the  actual  values  of 
N,  N',  and  /at  intervals  during  the  entire  period  for  which  the  per- 
turbations are  required,  and  to  interpolate  their  values  for  the  inter- 
mediate dates.  The  variations  of  these  quantities  arising  from  the 
variations  of  SI,  i,  SI ',  and  i'  may  also  be  determiiied  by  means  of 
differential  formula}.  Thus,  from  the  difterential  relations  of  the 
parts  of  the  spherical  triangle  from  which  the  equations  (130)  are 
devived,  we  easily  find 


VAUfATION    OF   C()-01MHNATKS. 


1J»l 


j,r>  Mm  ,.     ,,^,  _.  Hill  iV  ,    ,.,     ,      sill  .V    ,. 

dN' =  .    ,co8iV  rf(R  —  ft)—    .    ,cort/(/t'4-    .    ,  di, 
mix  I  sill  /  bill  J 

J.,       Hint'        „,  ,._,       _,      sill  .V  ...   ,    siiiiV        ,  ,.    (136) 

fill  /  »  w  ^m  y  jjllj   ^ 

rf/    =cusiV'(/t' —  COS  iVf/t  + sin  ('.sill  iVr/(  Q' —  Ji). 
When  /  anil  /are  very  small,  it  will  bo  hotter  to  use 


Hin ' 


sill  .V 


sill  ( 


!*iii  A'' 


sin/       sin  (ft'  — Si)'  sin/    "  8in(  ft' —  ft  )' 


(137) 


in  findiiif^  tlio  niiinorical  valiies  nf  these  cooflicieiits.  JJy  moans  of 
those  f'ornuihe  we  may  derive  the  values  of  fh\\  oN'y  and  o/  corre- 
spandinp  to  given  values  of  oft,  di,  ^ft',  and  <)/'.  The  formuho 
by  niean.s  of  wliioh  o(T,  fJft,  and  di  may  be  obtained  direetly,  will  be 
lirosontly  eonsidered. 

The  results  for  iiX,  dN',  and  fJ/ being  applied  to  the  quantities  to 
whieli  they  belong,  we  may  eomiiute  the  aetuul  values  of  ir'  and  ;5'. 
The  val  !  of  r  will  be  found  from  the  given  value  of  v,  and  that  of 
w  will  be  given  by  means  of  equation  (l.'Jo).  Then,  by  moans  of 
the  formuhe  (132),  the  forces  li,  S,  and  Z  will  be  obtained.  The 
porturbataons  will  first  be  computed  in  reference  only  to  terms  do- 
ponding  on  the  first  power  of  the  disturbing  force,  and,  whenever  it 
hooomes  necessary  to  consider  the  terms  of  the  .second  order,  the 
rosults  already  obtained  will  enable  us  to  estimate  the  valu(>s  of  the 
jwrturbations  for  two  or  more  intervals  in  advance  with  sutHcient 
accuracy  for  the  determination  of  the  three  required  components  of 
the  disturbing  force;  and  when  there  are  two  or  more  disturbing 
bodies  to  be  considered,  the  forces  for  each  of  these  may  be  comp.  jd 
at  once,  and  the  values  of  each  component  for  the  several  disturbing 
bodies  may  be  united  into  a  single  sum,  thus  using  I'R,  1\S,  and  I'Z 
in  place  of  li,  S,  and  Z  respectively.  The  approximate  values  of  the 
porturbations  will  also  facilitate  the  indirect  calculation  in  the  deter- 
mination of  the  complete  values  of  the  required  diirerential  coeffi- 
cients. 

183.  When  only  the  perturbations  due  to  the  first  power  of  the 
disturbing  force  are  required,  the  osculating  elements  ft^  and  i^  will 
be  used  in  finding  N,  N',  and  I,  and  r„,  tOg  will  be  used  instead  of  r 
and  ?»  in  the  calculation  of  the  values  of  li,  S,  and  Z.  The  equations 
for  the  determination  of  the  perturbations  dM,  v,  and  dz„  neglecting 
terms  of  the  second  order,  are,  according  to  the  equations  (110), 
(115),  and  (129),  the  following:— 


m 


I    y 


492 


THEORETICAL   ASTRONOMY. 


*l'';'o(l+»n) 


(138) 
kHl-^m) 


/fe'Cl+m), 


Tlio  valuo  of  V  is  first  foinul  by  intcj^ration  from  tlm  results  jjivon 
by  tlio  socoiul  of  tlu'so  equations,  auil  ihcji  JJf  is  found  from  the  first 
C(iuation.  Finally,  Sz,  is  found  l)y  means  of  the  last  e(|nation.  The 
intej^rals  are  in  eaeli  ease  o(|ual  to  zero  for  the  dates  to  which  the 
fundamental  oseulating  elements  belonj;,  and  the  process  of  inte<;ru- 
tion  is  analogous,  in  all  respeet*",  to  that  already  illustrated  in  the 
case  of  the  variation  of  the  rectangular  co-ordinates.     It  will  be  ob- 

8erve<l,  however,  that  the  expression  for  .  involves  only  one  indi- 
rect terra,  the  coefficient  of  which  is  small,  and  the  same  is  true  in 


the  case  of  --j^i  while  — ,'^  is  given  directly.  When  the  perturba- 
tions have  been  found  for  a  few  dates,  the  values  for  the  following 
date  can  l)e  estimated  so  closely  that  a  repetition  of  the  calculation 
will  rarely  or  never  be  rcfjuired;  and  the  a(!tual  value  of  /•  may  be 
used  instead  of  the  approxinmte  value  r^  in  these  expressions  for  the 
diflerential  coefficients.  Neglecting  terms  of  the  second  order,  we 
have 

logr  =  logr, -f- V> 

wherein  Xq  denotes  the  modulus  of  the  system  of  logarithms.  We 
may  also  use  v,  instead  of  v^,;  but  in  this  case,  since  r,  and  v,  depend 
on  dM,  only  the  quantities  required  for  two  or  three  places  may  be 
computed  in  advance  of  the  integration. 

A  comparison  of  the  equations  (138)  v/itli  the  complete  equations 
(110),  (115),  and  (129)  shows  that,  if  the  values  of  /3'  and  w'  are 
known  to  a  sufficient  degree  of  approximation,  we  may,  with  very 
little  additional  labor,  consider  the  terms  depending  on  the  squares 
and  higher  powers  of  the  masses.  It  will,  however,  appear  from 
what  follows,  that  when  we  consider  the  perturbations  due  to  the 
higher  powers  of  the  disturbing  forces,  the  consideration  of  the  effect 
of  the  variation  of  z,  in  the  determination  of  the  heliocentric  place 
of  the  disturbed  body,  becomes  much  more  difficult  than  when  the 
terms  of  the  second  order  are  neglected ;  and  hence  it  will  be  found 
advisable  to  determine  new  osculating  elements  whenever  the  con- 
sideration of  these  terras  becomes  troublesorae. 


VARIATION   OK   «  O-OUIUXATI-X 


493 


Tlio  results  may  l)o  convcincntly  oxpiN'ssod  in  seconds  of  arc,  and 
afd'rwanis  v  and  th,  may  Ix;  coiivi'rtcd  into  tlicir  values  (expressed  in 
units  «»('  tlio  .seven til  decimal  place,  or,  >;ivin^?  proper  attention  to  the 
lutinojjeneity  of  the  sevi-ral  terms  of  the  (Mpiations,  in  the  nnmeri<-al 
operations,  rM/"  may  he  expressed  in  seconds  of  are,  while  u  and  ih, 
are  oi)tained  directly  in  units  of  the  seventh  decimal  place.  It  will 
he  advisable,  also,  to  introduce  the  interval  lo  into  the  tormuhe  in 
Hiich  a  ntanner  that  this  (piantity  may  be  omitted  in  the  ease  of  the 
liirmuhe  of  integration. 

184.  In  the  case  of  orbits  of  pjreat  eccentricity,  the  mean  anomaly 
and  the  mean  <lailv  motion  cjinnot  be  convenientlv  used  in  the  nu- 
nierical  application  of  the  formula'.  Instead  of  those  we  must 
employ  the  tinu;  of  perihelion  [lassaj^e  anil  the  elements  7  and  c. 
Thus,  let  jfy  be  the  time  of  perihelion  passage  for  the  osculat'ug  ele- 
ments for  the  date  /„,  and  let  ?{,  +  "J'be  the  time  of  perilndion  pas- 
Kige  to  be  used  in  the  formulie  in  the  place  of  T^  and  in  coi.neetion 
with  the  elements  </„  and  c^  in  the  determination  of  t'lo  values  of  /•, 
and  v„  so  that  we  have 

In  the  case  o?  parabolic  motion  wo  have,  neglecting  tlie  mass  of  the 
dis-turbed  botly. 


V2  9oii 


=  tan  ]v,  +  J  tau'  .Jv„ 


(1.39) 


the  solution  of  which  to  find  t\  is  effected  by  means  of  Table  VI.  as 
ah'oady  explaine*!.     To  find  >•„  we  have 


r,  —  7o  sec'  h,. 


For  the  other  cases  in  wliich  the  elements  M^  and  /i^^  cannot  be  em- 
ployed, the  solution  must  be  offeete<l  by  means  of  Table  IX.  or  Table 
X.    Thus,  when  Table  IX.  is  used,  we  compute  M  from 


wherein  log  CJ,  =  9.9601277,  and  with  this  as  the  argument  we  derive 

from  Table  VI.  the  corresponding  value  of  V.     Then,  having  found 

l  —  e 
t  =  ^       ",  by  means  of  Table  iy<.  wc  derive  the  coefficients  required 

in  the  equation 

r,  =  F  4-  ^  (1000  +  B  (1000'  +  C(1000»,  (140) 


n 


494  THEORETICAL   ASTRONOMY. 

from  which  v,  will  be  determinecl.     Finally,  /•,  will  be  fouiul  from 


1  -f-  t'o  COS  V, 


(141) 


When  Ta1)lc  X.  is  used,  we  procecl  an  explained  in  Art.  41,  usini; 
the  cK'MU'nts  2'--  Tq-\-  32)  %aud  f'o, and  thus  we  obtain  the  required 
values  of  r,  and  »•,. 

It  is  evidiMit,  therclbre,  that,  for  the  determination  of  the  pertur- 
bations, oidy  the  formula  for  finriufj;  the  value  of  dJr  requires  modi- 
fication in  the  ease  of  orbits  of  great  eccentricity,  and  this  modilica- 
tion  is  easily  effected.     The  expression 


gives 


or,  simply, 


and  the  equation  (110)  becomes 


f^a'^T, 


(It 


=  1 


1 


Cl-fO'     (1+")'  kV^iH^i) 


-7^--.=-   fSrdf, 


(142) 


by  means  of  wiiich  the  value  3T  required  in  the  sohuion  of  the  equa- 
tions for  V,  and  v,  may  be  found. 

If  we  denote  by  t,  the  time  for  whi(!h  the  true  anomaly  and  the 
radius-vector  computed  by  means  of  the  fundamental  osculatina;  ele- 
ments have  the  values  which  have  been  designated  by  v,  and  r„  ro 
spectivcly,  we  have 

1    dm      dt^ 
dt' 


and  th6  equation  (110)  becomes 


1  + 


/'o 


dt 


dt, 

dt 


^l  + 


(!  +  .)•'  '   (!  +  >')'    kVp, 
or,  putting  t,='t-\-  dt, 


dt, 


(143) 


dt 


(lH-v)> 


-1  + 


(1  +  ")'    kVp„{l+m) 


/■ 


SrdL       (144) 


If  we  determine  8t  by  means  of  this  equation,  the  values  of  the 
radius-vector  and  true  anomaly  will  be  found  for  the  time  t-\-ot 
instead  of  /,  according  to  the  methods  for  the  diflereut  conic  sections, 


VARIATION   OF  CO-ORDINATES. 


495 


usiiip;  the  fundamental  osculatinpj  elements.   The  results  thus  obtained 
arc  the  required  values  of  v,  and  i',  respeetively. 

18').  When  the  values  of  the  perturbations  v,  8z„  and  (iM,  ilT,  or 
()/  have  been  determined,  it  reniains  to  liud  the  place  of  the  disturbed 
body.     The  heliocentric  longitude  and  latitude  will  be  given  by 


cos  b cos (1  — 

Sl)- 

r=cos(A—  Q), 

cos  b  sin  (1  — 

Sl)-- 

=  sin  (k  —  SI)  cos i, 

sin  6 

=  sin  (A  —  ^)  sin  i, 

or,  since  ^  ■==  ^,  —  <t  +  SI, 


cos  b  cos  (I  —  ft  )  =^  cos  (l,  —  (t), 
cos  b  sin  {I  —  ft  )  ^^  sin  {a,  —  o-)  cos  i, 
sin  6  ---  sin  (A,  —  a)  sin  /, 


(145) 


in  which  X,  =  v,-\-7rQ.  If  we  multiply  the  first  of  these  equations 
by  cos  (ft  — /i),  and  the  second  by  —sin  (ft  — h),  in  which  h  may 
have  any  vnlue  whatever,  and  add  the  results;  then  multiply  the  first 
by  (sin  ft  —  h),  and  the  second  by  cos  ( ft  —  h),  and  .  h\,  we  get 

cos  b  cos  (l—h)==co!i (A, — a)  cos  ( ft  — /i)— sin  (X, — a)  sin  ( ft  —h)  cos  i, 
cos  b  sin  (/ — h)^=^coii  (A, — (t)  sin  ( ft  — A)+siu  (A, — j)  cos  ( ft  — h)  cos  ?, 
sin  6  =sin  (A, — <t)  sin  i. 

Rut,  since  ^,  —  <t  =  (^,  —  ft^)  —  («■  —  ft„),  these  equations  may  be 
written 

cos  6  cos  (^  —  h) 

=cos(A, — ft„)(cos(o-— ft^^cosCft— /(.)-}-sin((T— ftn)sin(ft— /i.)cost) 
-|-sin  ( A,— fto)  (sin  (o-—  ft „)  cos  (  ft  — /i)— cos  (t— ft „)  sin  { ft  —h)  cos  i), 

cosisinC?  — /i)  (14G) 

=cos  (A, — ft,)  (cos  (ff— ft „)  sin  ( ft  — /<)  — sin  (<r— ft „)  cos  (  ft  —h)  cos  i) 
+sin  'A,— fto)(sin  f<J— fto)sin(  {,g— /i.)4-cos(rr— ft„)cos(ft  — /i)co3('), 

sin/>— sin(A, — ft„}cos(<T— ftp)sini— cos(A, — ft  J  sin  (<t — ft^)  sin  i. 

Lot  us  now  conceive  a  spherical  triangle  to  be  formed,  of  which  two 
of  the  si  les  are  o  —  ft^  and  ft  --  h,  respectively,  and  let  the  angle 
included  by  these  sides  be  /.  Since  h  is  entirely  arbitrary,  we  may 
assign  to  it  a  value  such  that  the  other  angle  adjacent  to  the  side 
(^  —  fto  will  be  equal  to  i^.  Let  the  third  side  be  designated  by 
K  "'  fto>  ^^^  t'*c  angle  oj)posite  to  (r  —  ft„  by  jy'.  The  auxiliary 
triangle  thus  formed  gives  the  Ibllowing  relations : — 


496 


THEORETICAL   ASTRONOMY. 


cos(/(o— S^o)— cos(t— SJo)co8(Ji— /0+sin(<T— S2o)sin(Si— /i)cos?, 
sin  (/io — S^o^'^'"  '"o=sin  {Q — h)  sin  i,  (147) 

sin  {ho — Slj  cos  /o=sin  {<t—Qo^  cos  ( SI  — h) — cos  (<r —  Q^)  sin  (SI — h)  cos  /, 
sin  (/lo — J^j)cosij'=cos(<r — J^o)sin  (SI — /i)— sin  (ff— SI  o)  cos  (Si— h)coi{. 

Combining  these  with  the  preceding  equations,  we  easily  derive 

cos  6cos  (^— /i)=cos  (A, — Slo)  cos  (h^ — fto)+sin  (^,—Slo)  ^in  (h^ — Slo)  cos  i^, 
cos 6 sin  (/—/() =sin  (^,—  Slo)oos(ho — Slo)  cosjj— cosC-i,— J^o)sin  (ho—Slo) 
+cos(A-Sa„)sin(/io-fto)(l+cosV)  (148) 

4-9in  (-1,—  Slo)  ((c"s  j— cos  ig)  cos  (Ao— JJo)+sin  (a—  SI o)  sin  ( SI  —h)  sin'  i), 
sin  6=sin  /„sin  (-i,— J^o)+(cos  (ff—Slo)  sin  i— sinijsin  (A,—  Slo) 
—  cos(>l,— JJo)sin((T — ^j)sini. 

Since  the  actiou  of  the  component  of  the  disturbing  force  perpen- 
dicular to  the  plane  of  the  disturbed  orbit  does  not  change  the  radius- 
vector,  Ave  have 

r  sin  6  =^  r  sin  %  sin  (A, — Slo)  +  ^^n 


(149) 


and  hence  the  last  of  these  equations  gives 

dz 

-l  =  s'm(X,  —  Slo)  (cos  (<T—Slo)  sin  i  —  sin  i^) 

r 

— cos  (A,  —  JJ„)  sin  (a  —  J^j)  sin  i. 

From  the  relation  of  the  parts  of  the  auxiliary  spherical  triangle,  we 

have 

sin  i  sin  (a  —  Sl^  =  sin  ij'  sin  (h^  —  Slo), 

sin  i  cos  (a  —  J^^)  =  sin  jj'  cos  (h^  —  Slo)  cos  %  +  cos  jj'  sin  %. 


Therefore, 

Sz,        .    ,. 
— ^  =  sm  (/,  • 
r 


and 


Slo)  (cos  ?o  cos  (ho  —  Slo)  sin  i?'  —  sin  io(l  —  cos  rj')), 
—  cos  (X,  —  Slo)  sin  (ho  —Slo)  sin  rj', 


(150) 


sni  tj 


cost? 


7  =  sin(A,— Jio)(cos?oCOs(Ao — J^o)(l-f-cosi?')~sin?nsin)j') 

(151) 


—  cos  (A,— ^o)  sin  (/jo— Sio)  (1  +  cos  v). 
We  have,  further,  from  the  auxiliary  spherical  triangle, 
cos  i  =  sin  ^  sin  -q'  cos  (li^  —  J^o)  —  cos  ^  cos  Vi 
from  which  we  get 

cos  i  —  cos  ?o  =  sin  io  cos  (ho  —  Slo)  sin  ij'  —  cos  »o  (1  +  cos  r^'), 

We  have,  also, 

8in(«T —  S^o)sini=sinij'8in(Ao — Qo)» 
sin  (  ft  —  /t)  sin  I = sin  t'o  sin  (ho —  Q  o), 


VARIATION  OF  CO-ORDINATES. 


497 


or 

sin  (t  —  Qo)  sin  (  J2  —  h)  sin'  i  =  sin'  (/t«  —  ft  „)  sin  4  sin  v)'. 

Hence  we  derive 

(cosi— cosio)  cos  C^o— fto)+sin  (ff— fto)  sin  (Q—h)  sin't=sin4sin  V 

—  (I  +  COSV)  cos  ioCOS  (V-fto). 

Combining  this  and  the  equation  (151)  with  the  equations  (148),  we 
obtain 

cos  b  COS  (;-A)=cos  (A,-  ft  o)  COS  (Ao-  ft „) +sin(;,-  ft  „)  sin  (ho~  ft  „)  cos »;, 
cos  b  sin  il-h)=sm  (>l  —  ft  „)  cos  ( ^o-  ft  „)  cos  ^-cos  (X,—  ft  „)  sin  {h,— ft  „) 

sinij'         dz, 


smb 


Sz 

=sm(^,— ft,)sin  /q  H ^• 

r 


cos  T)'     r 


If  we  multiply  the  first  of  these  equations  by  cos(A„  —  ft„),  and  the 
second  by  — sin  (Ag— ft^,),  and  add  the  results;  then  multiply  the 
first  by  sin  (/to  —  fto),  and  the  second  by  cos(A„  —  fto),  and  add,  we  get 

cos6  cos(;-fto— (A-/0)-=cos(/l,-fto)+sin(;i„-fto)  ,-^^^  •  ^, 

1 — COS);      r  ' 

cos6  sin  (^-fto— (/t-/io))=i>in  (-t  —  fto)cos4-cos(Ao— fto)    ®'" '''       ''^' 

dz, 


smb 

Let  us  now  put 


=sin  (-*,— fto)  sin  4+- 


p'  —  s\n(a  —  fto)  sin  i, 

^  =  cos  ((T  —  fto)  sin  i  —  sin  io, 


1 — COS);'     r  ' 
(152) 


and  there  results,  from  (149), 


Sz, 

--  =  q'  sin  (X,  —  fto)  —p'  COS  (A,  —  fto). 

Comparing  this  with  equation  (150),  we  observe  that 

/  =  sin)?' sin  (/to— fto), 

q'  =  sin  1)'  COS  (/to  —  fto)  COS  io  —  sin  %  (1  —  cos  yj'). 

Therefore,  we  have 


(153) 


(154) 


sm); 


1  —  C08)J 


7sin(^o— fto) 


P 


COS)? 


sm)?            ,,            s                                </ 
i^ZTT;:;^  cos  (/to  —  fto)  =  tan  to  H —^ rs 

1  —  COS)?  COSlo(l — COS)?) 

83 


'\» 


498  THEORETICAL   ASTRONOMY. 

and,  if  wc  put  r=^h  —  /jq,  the  equations  (152)  become 


cos  b  cos  {I — S^o — /')=cos(A, —  Qo)+ 


P 


1  —  cosiy'     r  ' 
cosfisin  (I — Slo — ^')=sin  (^, — SJo)cosio— I  tani'o-} 


COSio(l 


(155j 

t \^ 

—  cos  r/)  f   r  ' 


sin  6 


=sin  (^f—Slo) sin  /o+- 


As  soon  as  P,  p',  q',  and  :y'  are  known,  these  equations  will  furnish 
the  exact  values  of  I  and  b,  those  of  X,  and  r  being  found  by  means 
of  the  perturbations  v  and  3M. 

186.  The  value  of  F  may  be  expressed  in  terms  of  j)'  and  q'. 
Thus,  if  we  differentiate  the  first  of  equations  (147)  and  reduce  by 
means  of  the  remaining  equations  of  the  same  group,  we  get 

dihg —  So)==cos'j'd(Ji  — h)  +  cosio(Z<T-f  sinijsin((r —  S^a^^h 

and  if  we  interchange  SI  — h  and  h^ —  Slo'^^  this  equation,  we  inust 
also  interchange  i  and  i^,  which  are  the  angles  opposite  to  these  sides, 
respectively,  in  the  auxiliary  spherical  triangle,  so  that  we  shall  have 

d{Q  —  h)  =  cos  rj'  d  (/lo  —  JJJ  +  cos  i  da, 

Iq  being  constant.  Adding  these  equations,  observing  that  Sl^h  also 
constant,  we  get 

(1 — cos5j')rf(Si— /i+A,)=sinroSin((r — £1^ di-\-{cosi-\-cosi^ da ',  (156) 

and  since  da  =  cos  idSl,  this  becomes 


{1— COST)')  d{h  —  h,) 
which,  since 


sin  ig  sin  (a  —  J^  J  di 

+  (sin'i  —  cosTj'  —  cosi  cos?o) 


da 
cos  i' 


(157) 


cos  ij'  =  sin  *  sin  i^  cos  (a  —  Sio)  —  cos  i  cos  io, 

may  be  written 

(1 — eosrj')dr= — 8iniosin(<r — Sio)di-\-taJii{sini — sin?"ocos((r  — Sio))^"- 

(158) 
The  differentiation  of  the  equations  (163)  gives 

dj)'  =  sin  (a  —  ^o)  cos  idi  -\-  sin  i  cos  {a  —  ^o)  da, 
dq'  =  cos  (a  —  Slo)  cos  idi  —  sin  i  sin  (ff  —  Slo)  da, 

from  which  we  derive 


VARIATION  OF  CO-ORDINATES. 


499 


^dp'  — p'd(f  =  sin*  idff  —  sin  /o  dp' 

=  cos  i  ( — sin  I'o  sin  (a —  JJ o)  di-\~tan  i (sin  i — sin  j'o cos  (<r —  ^ o))  da^ . 

Combining  this  with  equation  (158),  we  get 

cos  i  (1  —  cos  V)  dr  =  q'dp'  — p'dq', 


and  hence 


C0St(l — COSTj) 


(159) 


the  integral  being  equal  to  zero  for  the  instant  to  which  tlie  funda- 
mental osculating  elements  belong.  It  is  evident  from  the  equations 
(153)  that  p'  and  q'  are  of  the  ox'der  of  the  first  power  of  the  dis- 
turbing forces,  and  hence,  since  r/  diifers  but  little  from  180° — (i-f^)), 
it  follows  that,  so  long  as  i  is  not  very  large,  Fifi  at  least  of  tho 
second  order. 
The  last  of  equations  (145)  gives 


and  since 
this  becomes 


2,  =  r  sin  i  sin  A,  cos  a  —  r  sin  i  cos  A,  sin  ff, 


x  =  rcos  X,,  y^=r  sin  A„ 

3,  =  —  a;  sin  i  sin  <r  -l"  2/  ^^^  *  cos  <t. 
Comparing  this  with  equation  (116),  it  appears  that 

o  =  —  sin  i  sin  ff,  /?  =  sin  i  cos  <r, 

and  hence,  by  means  of  (153),  we  derive 


(160) 


and  also 


y  =  —  o  cos  Sio  —  /9  sin  Slot 

q'  =  —  a  sin  Slo  +  /9  cos  JJ„  —  sin  i^, 


dp'  r^    ^<*  •     r> 

-ji  =  -cosft„^-sma 


dl 
dt 


da 


0  dt' 
d,3 


(161) 


3in$^„^  +  cosJi,^^ 


From  the  equations  (118)  and  (121),  observing  that 


X 


dy 


dx 


2'-:77  =  *^^.P(i  +  "^)' 


dt      "  dt 

we  derive,  by  elimination, 

da, r  sin  ^,  cos  i   „  d£^ r  cos  ^,  cos  i   „ 

di~      kVpil  +m)    '  di       kVpUr^'m) 


500  THEORETICAL  ASTRONOMY. 

Therefore  we  shall  have 

rcosisinC^,  —  ft„) 


dp^ 

dt  ~      kV'pil  -f-m) 

d<f r  cos  i  cos  (^,  —  SIq) 

__  _ 


Z, 


Z, 


(162) 


kVp{l  +  w) 

by  means  of  which  p'  and  q'  may  be  found  by  integration,  the  inte- 
gral in  each  case  being  zero  for  the  date  tg  at  which  the  determina- 
tion of  tlie  perturbations  begins. 

When  the  yalue  of  dz,  has  already  been  found  by  means  of  the 
equation  (129),  if  we  compute  the  value  of  q',  that  of  p'  will  be 
given  by  means  of  (154),  or 

5z, 


jp--^'tana-ft„)--^^^^^_ 

and  if  p'  is  determined,  g'  will  be  given  by 

Sz, 


fto)' 


q'=p'cotiX,-Sl)-\- 


r  sin  (A,  —  Slo)' 


If  both  p'  and  q'  are  found  from  the  equations  (162),  dz,  may  be  de- 
termined dii'ectly  from  (154);  but  the  value  thus  obtained  will  be 
less  accurate  than  that  derived  by  means  of  equation  (129). 

Since  the  formula  for  -^  completely  determines  the  perturbations 

due  to  the  action  of  the  component  Z  perpendicular  to  the  plane  of  the 
instantaneous  orbit,  instead  of  determining  jj'  and  q'  by  ar  independent 
integration  by  means  of  the  results  given  by  the  equations  (162),  it 

will  be  preferable  to  derive  them  directly  from  3z,  and  --^'-  The 
equations  (161)  give 


p'  —  —  cos  Q„  <5a  —  sin  Slo  ^A 


g'  =  —  sin  Slo  ^^  +  cos  £1^  «/?• 


Substituting  for  da  and  d^  their  values  given  by  (125)  and  (126), 
and  putting 


x"  =  a;  cos  Jio  +  2/  sin  Sl^^ 
we  obtain 


f=:  —  xsmSlo  +  y  cos  Slo> 


hVp  (1  +  m 
1 


kVp{\  +m 


I      ddz,  df\ 

)V    dt      ^''  dty 


(163) 


VARIATION  OF  CO-ORDINATES. 

Substituting  further  the  vahies 

'  ^  r  cos  {^,  —  Slo), 


501 


and  also 


y"  =  r  sin  {X,  —  J^,), 


dt 
dr 

lit 


kVl+m 


Vp 


e  sin  V  ■ 


kVpa  +  m) 


esin  V 


1  -j-  e  cos  V 


we  easily  find,  since  X,  —  'W  =^  >f> 

^'  =^  -  (cos  (-1,  —  S2o)  +  e  COS  (;if  -  ^„))  -^  +  — /.y.  --f -"-'^-  •  7,. '. 
5'  =  +  (sin  (;,  -  Ji„)  +  e  sin  {x  —  fto))  ~  +  TT 7-==T  '  V' 


(164) 


which  may  be  used  for  the  determination  of  p'  and  q'.  These  equa- 
tions require,  for  their  exact  solution,  t..«,t  the  disturbed  values  c,  ;f, 
and  JO  shall  be  known,  but  it  is  evident  that  the  error  will  be  slight, 
especially  when  e  is  small,  if  we  use  the  undisturbed  values  f^,,  ^Ju, 
and  ;fu  =  ttq.  The  actual  values  of  ^  and  r  are  obtained  directly  from 
the  values  of  the  perturbations. 

When  p'  and  q'  have  been  found,  it  remains  only  to  find  cos  /,  and 
1  —  cos  r/,  in  order  to  be  able  to  obtain  F  by  means  of  the  ecjuation 
(159).     From  (153)  we  get 


and  hence 


pit  _|_  g/j  __  gjjjj  I  —  g;jj2 1^  —  2g'  sin  \, 


COSl: 


i/r= 


^■^-(3'  +  sini„n 


from  which  cosi  may  be  found.     The  equation  (157)  gives 

1  —  cos  5j'  =  COS  \  (cos  \  -\-  COS  i)  —  cl  sin  i„, 
by  means  of  which  the  value  of  1  —  cos  r/  will  be  obtained. 


(165) 
(166) 


If  we  substitute  the  values  of  p',  q',  -^i  and  -~  given  by  the 


dt    "        dt 
equations  (153)  and  (162)  in  (159),  it  is  easily  reduced  to 

Sz, 


^-Skz 


(1  —  cosV)  kVp{\  -\-  m) 


Zdt, 


(167J) 


v/hich  may  be  used  for  the  determination  of  P.  When  we  neglect 
terms  of  the  order  of  the  cube  of  the  disturbing  force,  in  finding  F 
we  may  use  po  in  place  of  p  and  put  1  —  cos  jy'  =  2  cos^  Iq,  so  that  the 
formula  becomes 


502 


THEORETICAL   ASTRONOMY. 


r= 


2cos'<;^V/;)o(l  +  '«) 


S"'' 


Zdt. 


(168) 


187.  By  means  of  tlie  formulae  which  have  tlius  been  derived,  we 
may  find  the  values  of  all  the  quantities  recjuired  in  the  solution  of 
the  equations  (155),  in  order  to  obtain  the  values  of  /  and  b  for  the 
disturbed  motion.  From  r,  /,  and  6  the  corresponding  geocentric 
place  may  be  found.  The  heliocentric  longitude  and  latitude  may 
also  be  determined  directly  by  means  of  the  equations  (145),  provided 
that  Q,,  <T,  and  i  are  known;  and  the  required  formuUe  for  the  deter- 
mination of  these  elements  may  be  readily  derived.  Thus,  the  equa- 
tions (160)  give,  by  differentiation, 


v;hence 


do.             .            .  di 

- ,-  —  —  sm  (T  cos  I  -j/  — 
dt                            dt 

-  sm  I  cos  ff  -rr, 

djS                         .  di 

-— -  =         cos  <r  cos  t  —rr  — 

dt                            dt 

...       dn 
■  sm  I  sm  <r  -rr, 

.     .  d<T                         da. 

sm  I  -rr  =  —  COS  (T  -  .-- 

dt                     dt 

.  di              .       da 

dH 

COSt-TT  = 

dt 


sm  a    ,-  +  cos  a 
dt 


dt' 


J  J  a 

Introducing  the  values  of    ,.  and  -J-  already  found  into  these  equa- 
tions, and  putting 


ff  =  ffj  +  (Jff  =  Ji„  -f  dff, 
we  obtain 


*  ==  ^  +  ^i, 


Sl  =  Slo  +  ^Sl, 


ddff  _ 1_ 

dt  ~  kl/p{l-\-m) 

d5i^ 1 

dt 


cot  i  sin  (A,  —  <r)  rZ, 


-  cos  (A,  —  ff)  rZ, 


(169) 


kVpiX+m) 
and  also,  since  d<r  =  cos  i  t?  J^ , 

dSQ  1  sin  (X,  —  a) 


dt 


kVp{l-{-vi)  sini 


rZ, 


(170) 


by  means  of  which  the  variations  of  <t,  i,  and  Si  due  to  the  action 
of  the  disturbing  forces,  may  be  determined.  The  integral  is  in  each 
case  equal  to  zero  at  the  initi  I  date  <o  to  which  the  fundamental  os- 
culating elements  belong  and  at  which  the  integration  is  to  coin- 
mence. 


VAUIATION   OF  CO-ORDINATES. 

If  we  find  /,  and  then  a  —  ft  from 

/•       tan  \i 


» 


<r  =  I  — r—  ^ " sin  {\,  —  <t)  yZiU, 


503 


(171) 


the  true  longitude  iu  the  orbit  will  he  obtained  from 


It  IS  evident  that  since  the  exi)ressions  tor  —r-f  -m  and 

"■  at     at  (It 


f/''ft 


re- 


(juire,  for  an  accurate  solution,  that  the  disturbed  values  /,  <t,  and  p 
siiail  be  known,  and  require,  besides,  that  three  separate  inte«f  rat  ictus 
shall  be  performed,  unless  the  perturbations  are  computed  only  in 
reference  to  the  first  power  of  the  disturbing  force,  in  wliich  case  we 
use  /'„,  Pq,  and  ft^  in  place  of  i,  p,  and  a,  respectively,  in  the  C([uations 
(169)  and  (170),  the  action  of  the  comi)onent  Z  aim  be  considered  in 
the  most  advantageous  manner  by  means  of  the  variation  of  z,  arising 
from  this  component  alone;  and  even  when  only  the  perturbations 
of  the  first  order  are  to  be  determined  it  will  still  be  preferable  to 


d''lz, 


,  and  to 


derive  dz,  by  the  indirect  process  from  the  expression  for     , 

determine  the  heliocentric  place  by  means  of  the  equations  (lo5). 
Wlien  we  neglect  the  terms  of  the  second  order,  these  equations 
become 

cos  6  cos  (^  —  fto)  =  cos(-l, —  fto), 


cos  b  sin  {I  —  ft  o)  —  siu  (x,  —  ft  „)  cos  i^  —  tan  i^  —^, 


(172) 


sin  6 


sin  (A,—  fto)sinio+  — -, 


by  means  of  which  I  and  b  are  determined  immediately  from  the  per- 
turbations (IM,  V,  and  oz,.  The  peculiar  advantage  of  determining 
the  eft'ect  of  the  action  of  the  component  Z  by  means  of  the  partial 
variation  of  s,  is  apparent  when  we  observe  that  the  expressions  for 

-rr  and  — r—  involve  sin  i  as  a  divisor;  and  in  the  case  of  orbits  whose 
at  (it 

inclination  is  small,  this  divisor  may  be  the  source  of  ^  consideral)le 
amount  of  error. 

188.  The  determination  of  the  perturbations  so  as  to  include  tiie 
higher  powers  of  the  masses  is  readily  effected  by  means  of  the  com- 
plete expressions  for  —j—,  ^.y.  and  -jt^>  when  the  correct  values  of 
R,  Sf  Z,  if  and  p  are  known.     The  corrected  values  of  /  and  p — 


504 


THEORETICAL   ASTRONOMY. 


which  are  required  only  in  the  case  of  8z, — may  he  easily  estimated 
with  sulliflent  afcuraoy,  since  wo  reciuirc  only  cos/,  while  V p  ap- 
j)cars  as  the  divisor  of  a  term  whose  numerical  value  is  gcnoraliy 
insignificant.  To  obt^iin  the  actual  values  of  li,  >S',  and  %,  the  cor- 
rections to  be  applied  to  N,  iV',  and  /must  first  be  determined  by 
means  of  the  formulte  (13(J).  The  values  of  di'  and  tlQ'  will  be 
found  by  meaiiK  of  the  data  furnished  by  the  tables  of  the  motion  of 
the  disturbing  l)ody,  an<l  the  corresponding  corrections  for  N,  N', 
and  /  having  been  found  by  means  of  the  terms  of  (136)  involving 
di'  and  dSl',  there  remain  the  corrections  due  to  ol  and  oSl  to  be 
applied.  These  may  be  found  in  terms  of  the  quantities  p'  and  q' 
already  introduced.     Thus,  the  ecjuations 


dp'  =  cos  i  sin  (it  —  JJ^)  di  -\-  sin  i  cos  (a 
dq'  =  cos  i  cos  {a  —  Sio)  <^*  —  **•"  *  ^^^  '-f 


give 


cos  i  di  =  sin  (t 
sin  /  dff  =  cos  («r 


0,9)  dj/  —  sin  (a 


Sio)d<T, 
fto)fV. 


The  equations  (136)  give,  observing  that  da  =  cos  i  dSl, 

dl    =  —  cos  N  di  —  tan  i  sin  iV  da, 

,--.,        ,    siuiV  ,.        tani         „, 

dN  =  +  -•  — r  dt . — J-  cos  Nda, 

sui  I  sm  i 

and,  substituting  the  preceding  values  of  di  and  dff,  these  become 


di  ^  -  '^'1^l±j^=lRo1  dp'  -  ""^ ^^'  tar  ^°^  ^g^ 


cos  t 


cos  I 


dN'  =:  -  gg^-t"--^  dp'  +  '-^-^in  ^-  d<l'- 


sin /cos  i 


sin /cost 


If  we  neglect  the  perturbations  of  the  third  order,  these  equations 
give 


dl    =  — siniV-— - — cosiV-^^, 


cos  In 


COSVo 


3N'  =  —  cosec  /(  cos  N-^  —  sin  N  - 


COStj 


COS 


l)- 


(173) 


by  means  of  which  81  and  dN  may  be  determined,  p'  and  q'  being 
found  by  means  of  the  equations  (164),  using  c^,  ^r^,  and  />o  i"  p'li^^s 
of  e,  ■)(,  and  p.  The  results  for  81  and  8N'  obtained  from  (173) 
being  applied  to  the  values  of  /'  and  iV'  as  already  corrected  on 
account  of  8i'  and  8^',  give  the  requii'ed  values  of  these  quantities. 


mm 


NUMERICAL   EXAMPLE. 

When  wc  considor  only  di  nnd  dQ ,  since 

sin  i'  cos  N'  =  cos  i  sin  i  +  sin  i  cos  /  cos  N, 

we  easily  find 

^iV=cos/<5iV'  — 5<r, 


506 


(174) 


and  if  wo  add  the  quantity  coi^IdN'  to  the  value  of  N  already  cox*- 
rected  on  account  of  di'  and  dSi',  and  denote  the  result  by  iV„  the 
required  value  of  N  will  he  N,  —  d(T.  Then,  according  to  (131),  we 
may  compute  lo'  +  da  and  ,^'  by  means  of  the  fornmlie 


tan  ((u''  -j-  (5(r)  —  N,)  =  tan  v^'  cos  7, 
tan  (*'  —  tan  /sin  ((w'  +  <5«t)  —  N,), 


(175) 


using  the  values  of  N'  and  /  as  finally  corrected.  We  have,  further, 
according  to  (135), 

by  means  of  which  we  may  compute  the  value  of  w  +  drr;  then  the 
value  of  vd'  —  ic  required  in  the  equations  (132),  and  also  in  finding 
the  value  of  p,  will  be  given  by 

w'  —  tt)  =  (it)'  -j-  da)  —  {w  +  dff), 

and  the  forces  i?,  S,  and  Z  may  be  accurately  determined. 

By  thus  determining  the  correct  values  of  R,  S,  and  Z  from  date 
to  date,  the  perturbations  SM,  v,  and  oz,  may  be  determined  in  refer- 
ence to  the  higher  powers  of  the  disturbing  forces  according  to  the 
process  already  explained.  The  only  difhculty  to  be  encountered  is 
that  which  arises  from  the  quantities  F,  p',  and  q',  required  in  the 
determination  of  the  heliocentric  place  of  the  disturbed  body  by 
means  of  the  equations  (155).  If  an  exact  ephemeris  for  a  short 
period  is  required,  by  means  of  the  complete  perturbations  we  may 
determine  new  osculating  elements,  and  by  means  of  these  the  required 
heliocentric  or  geocentric  places. 

189.  Example. — We  will  now  illustrate  the  application  of  the 
formulae  for  the  determination  of  the  perturbations  8M,  u,  and  8z,  by 
a  numerical  example;  and  for  this  purpose  let  it  be  required  to 
determine  the  perturbations  of  Eurynome  @  arising  from  the  action 
of  Jupiter   from   1864  Jan.  1.0  to   1865  Jan.  15.0,  Berlin   mean 


606 


TIIKOIIKTICAL   AHTUONOMY. 


timo,  tlio   fiindiiinentttl   osciiliiting  olcnifiitrt   being   those   given   in 
Art.  l(i(j. 

In  the  first  phico,  by  means  of  the  formuhe  (130),  using  the  values 


ft  =:  2(n\"  ;}fl'  r)".7, 

Q'=   98    58  22  .7, 


t*=4°  n()'52".l, 
i'=-l    18  40  .5, 


which  refer  to  the  ecliptic  and  mean  equinox  of  18(50.0,  we  obtain 

N==  194°  0'  49".i),       N'  ^  .301°  38'  31".7,       /  =  5°  9'  r>(>".4. 

Then,  I)y  means  of  the  data  furnislied  by  the  Tahlcfi  of  Jiijtlfcr,  \vc 
find  the  values  of  n',  the  argument  of  the  latitude  ol' Juj titer  in  reler- 
encc  to  the  ecliptic  of  18G0.0,  and  from  the  ecpiations  (131)  we  derive 
w'  and  /9'.  The  values  of  /•'  are  given  by  the  Tables  of  JujiUrr,  and 
the  values  of  ?•„  and  t'y  are  found  from  the  elements  given  in  Art. 
166.     The  results  thus  obtained  are  the  following: — 


lll'llill  MlHIl 

Tliiif. 

loRro 

'•o 

lOR  I-' 

IV' 

/3 

18r,;{  Dec. 

12.0, 

0.291084 

354= 

20' 

18".0 

0.73425 

14°  18'  .>t''.0 

—  0°  l'38".l 

18G4  Jan. 

21.0, 

0.29-48:{7 

10 

>> 

45 

.7 

0.73308 

17  21  44 

.2 

0  18  9 

.1 

Mnrd 

1  1.0, 

o.;?oor)74 

25 

24 

59 

.4 

0.73305 

20  25  5 

.2 

0  34  39 

.9 

April 

10.0, 

0.31 0SG4 

40 

13 

31 

.8 

0.73237 

23  28  59 

.8 

0  51  7 

.6 

»ray 

20.0, 

0.;?24298 

54 

14 

41 

.4 

0.73104 

20  33  32 

.1 

1  7  29 

.7 

.June 

29.0, 

0.3:59745 

07 

21 

23 

.5 

0.73080 

29  38  44 

.8 

1  23  4:$ 

.5 

Aug. 

8.0, 

0.350101 

79 

32 

18 

.1 

0.7.3003 

32  44  41 

.2 

1  39  40 

.3 

Sept. 

17.0, 

0..3724()9 

90 

49 

57 

.0 

0.72915 

35  51  24 

.6 

1  55  35 

2 

Oct. 

27.0, 

0.38S214 

101 

19 

9 

.8 

0.72823 

38  58  57 

.5 

2  11  7 

.5 

Dec. 

CO, 

0.402S94 

111 

5 

42 

2 

0.72720 

42  7  23 

.3 

2  20  20 

.3 

1805  Jan. 

15.0, 

0.410240 

120 

15 

32 

.0 

0.72025 

45  16  43 

.9  ■ 

-2  41  10 

.6 

The  value  of  w  for  each  date  is  now  found  from 


w- 


^^•o  +  ^o-«o  =  n■^197°38'6".5, 


and  the  components  of  the  disturbing  force  are  determined  by  mean.s 
of  the  fornuilaj  (132),  p  being  foiin.i  ^'rom  (133)  or  (134),  and  h  from 
(70).     The  adopted  value  of  the  mass  of  Jupiter  is 


m' 


1047.879 


and  the  results  for  the  components  E,  S,  and  Z  are  expressed  in  units 
of  the  seventh  decimal  j)lace.  The  factor  to^  is  introduced  for  conve- 
nience in  the  integration,  (o  being  the  interval  in  days  between  the 
successive  dates  for  which  the  forces  are  to  be  determined.  Thus  we 
obtain  the  following  results: — 


NUMEniCAL   EXAMPr.E, 


507 


Dutc. 

(-»« 

<..'.SV„ 

'.''/con  /„ 

(.jSr„iU 

1863  Dec. 

12.0, 

+ 

70.82 

4-     7.10 

+  0.04 

+■    1.37 

1864  Jan. 

21.0, 

68.05 

-    32.76 

0.49 

-11.45 

March 

1.0, 

61.16 

70.38 

0.92 

63.32 

April 

10.0, 

4«..")7 

102.01 

1.32 

150.48 

May 

20.0, 

32.77 

128.34 

1.68 

miio 

June 

2U.0, 

+ 

ir).41 

145.3U 

1.96 

404.35 

Au;r. 

«.(), 

2.1'J 

153.44 

2.17 

554.54 

Sept. 

17.0, 

10.12 

152.41 

2.29 

708.21 

Oct. 

27.0, 

34.81 

142.50 

2.25 

8.')»;.39 

Doc. 

6.0, 

48.95 

124.04 

2.0!) 

990.36 

180;')  Jan. 

15.0, 

61.45 

—    07.36 

+  1.75    - 

-1101.73 

0" 

V 

38".l 

0 

18 

9 

.1 

0 

M 

39 

.9 

0 

51 

/ 

.6 

1 

1 

29 

.1 

1 

2.3 

4:? 

.5 

1 

;«) 

4tJ 

.3 

1 

r^r, 

3.^1 

•) 

2 

11 

7 

.5 

2 

2(5 

20 

.3 

2 

41 

10 

.6 

Tlic  sijif^lo  integration  to  find  (oiSffjiU  is  efl'cotcd  by  mcan.s  of  tlie 
forniula  ('>2). 

Tlio  iHniations  for  the  ''"termination  of  the  required  differential 
coefticients  are 


dt 

dt 


4ih:rfi''''^'~^''')' 


0>'Ji 


2Mk' 


'  n  'a 


^•1 


1         C<    u      ''oS'n^'o    2  0 
,  -  til  I  lirM -  w'o  — 


,3     "' 


(P<h, 


ui^Z  COS  /„ 


(5» 

Substituting  in  these  the- results  already  obtained,  and  also 

log /x„  =  2.907809,        logi;o  =  0.371237,        log  e„  =  9.290776, 

wc  obtain  first,  by  an  indirect  process,  as  illustrated  in  the  case  of 

tlic  direct  determination  of  the  pertui'bations  of  the  rectangular  co- 

d'v  d''<h, 

ordinatcs,  the  values  of  iti^-ia  ^^'^^  '"^    /iir''  '^^^^  then,  liaving  found  v, 

10  '  is  given  directly  by  the  first  of  the.se  equations.  The  integra- 
tion of  tlie  results  thus  derived,  by  the  formula}  for  mechanical  quad- 
rature, furnishes  the  required  values  of  v,  8M,  and  dz,.  The  calcula- 
tion of  the  indirect  terras  in  the  determination  of  v  and  dz„  there 
being  but  one  such  term  in  each  case,  is,  on  account  of  the  smallness 
of  the  coefficient,  effected  with  very  great  facility. 
The  'final  results  are  the  following : —  ♦. 


508 


THEORETICAL  ASTRONOMY. 


I 


Date. 

1«G3  Deo. 

18(J4  Jan. 
!March 
April 
Slay 
June 
Aug. 
8ej)t. 
Oct. 
Dec. 

1865  Jan. 


12.0, 
21.0, 

1.0, 
10.0, 
20.0, 
29.0, 

8.0, 
17.0, 
27.0, 

6.0, 
15.0, 


■  0".028  + 
0  .072 

0  .499 

1  .213  + 

2  .070  — 

2  .902 

3  .546 
3  .858 
3  .723 
3  .056 

■1  .800  — 


36.16 
33.61 
22.55 
5.58 
13.52 
31.59 
4G.65 
57.88 
65.19 
68.83 
69.19 


"  di' 
+  0.04 
0.49 
0.89 
1.21 
1.45 
1.53 
1.60 
1.52 
1.28 
0.92 
+  0.40 


631 

+  0".01 

—  0  .01 

0  .27 


1  .11 

2  .75 
5  .24 
8  .49 

12  .22 

16  .05 

19  .49 

—21  .97 


+  4.41 

4.31 

37.11 

9i.f^6 

152.22 

199.05 

214.54 

183.69 

+  95.29 

—  58.00 

—279.84 


+  0.02 

0.04 

0.54 

1.93 

4.52 

8.54 

14.10 

21.24 

29.90 

39.82 

+50.64 


Since,  during  the  period  included  by  the.se  results,  the  perturbations 
ot'  the  second  order  are  insensible,  we  have,  for  the  perturbations  of 
Jiuri/iwme  arising  from  the  action  of  Jupiter  from  1864  Jan.  1.0  to 
1865  Jan.  16.0, 

8M=  —  21".97,        V  =  —  0.00002798,        Sz,  =  +  0.00000506. 

It  is  to  be  observed  that  d:,  is  not  the  complete  variation  of  the  co- 
ordinate z,  perpendicular  tc  the  ecliptic,  but  only  that  part  of  this 
variation  which  is  due  lO  the  action  of  the  component  Z alone;  am' 
hence  the  results  for  oz,  differ  from  the  complete  values  obtained 
when  we  compute  directly  the  variations  of  the  rectangular  co- 
ordinates. 

Let  us  now  determine  the  heliocentric  longitude  and  latitude  for 
1865  Jan.  15.0,  Berlin  mean  time,  including  the  perturbations  thus 
derived.     Frora  the  equations 

E,  —  e^  sin  E,  =  M„ 

r,   =Oo(l  — t'ocosi:,),  _ 

2  iv,  —  E,)  =  sin  A  ^„  sin  E,  J% 


sm 


we  obtain 


M,      =  99°  29'  35".51, 
log  r,  =  0.4162304, 
log  r  =  0.4162183, 


r  =  r,  (1  +  v), 

^,  =  110°  0'33".75, 
V,  -:  120  15  13  .80, 
X,  =  164  32  25  .97. 


The  c"'"ulation  of  the  values  of  r,  and  v,  from  the  values  of  J/„  <To, 
and  fj,  may  be  effected  by  means  of  the  various  formulae  for  the 


NUMERICAL   EXAMPLE. 


509 


4.41 

+  0.02 

4.31 

0.04 

7.11 

0.54 

i.d6 

1.93 

2.22 

4.52 

9.05 

8.54 

4.54 

14.10 

3.69 

21.24 

5.29 

29.90 

8.00 

39.82 

9.84 

+50.64 

)erturbations 

urbat 

ions  of 

[  Jan 

.  1.0  to 

detcrniination  of  the  radius-vector  and  true  anomaly  from  given 
elements.  If  we  substitute  the.se  results  for  ?.„  r,  and  dz,  in  th';  equa- 
tions (172),  we  get 


/  =  164°  37'  59".05, 


6  =  —  3°  5'  32".54, 


which  are  referred  to  the  ecliptic  and  mean  equinox  of  1860.0,  and 
from  these  we  may  derive  the  geoceiitrie  place  of  the  disturbed  body. 
If  the  place  of  the  body  is  required  in  reference  to  the  equinox  and 
ecliptic  of  any  other  date,  it  is  only  neces.sary  to  reduce  the  elements 
'o)  ^o>  '"^"^^  'o  *o  the  equinox  and  ecliptic  of  that  date;  and  then, 
having  computed  ?,,  and  r,  we  obtain  by  means  of  the  equations  (172) 
the  required  values  of  I  and  b.  In  the  determination  of  the  pertur- 
bations it  will  be  convenient  to  adopt  a  fixed  equinox  and  ecliptic 
throughout  the  calculation ;  and  afterwards,  when  the  heliocentric  or 
geocentric  places  are  determined,  tiie  proper  corrections  for  precession 
and  nutation  may  be  applied. 

In  order  to  compare  the  results  obtained  from  the  perturbations 
dMy  V,  and  dz,  Avith  those  derived  by  the  method  of  the  variation  of 
rectangular  co-ordinates,  we  have,  for  the  date  1865  Jan.  15.0, 

.ron=  — 2.5107584,        ;/„  = +  0.0897713,        z„  =  — 0.1400500 ; 
and  for  the  perturbations  of  these  co-ordinates  we  havo  found 

(Ja;  =  + 0.0001773.         J^/ :^  +  0.0001992,         J?  =^— 0.0000028. 
Hence  we  derive 

ar=r  — 2.5105811,         y^  +  0.6899705,         s  = —  0.1406618, 

and  from  these  the  corresponding  polar  co-ordinates,  namely, 

log  r  =  0.4162182,        I  =  164°  37'  59".05,         6  =  —  3°  5'  32".54, 

from  which  it  appears  that  the  agreement  of  the  results  obtained  by 
the  two  methods  is  complete. 

190.  When  the  perturbations  become  so  l.i.rge  that  the  terms  of  the 
E'cond  order  must  be  retained,  the  approximate  values  which  may  be 
obtained  for  several  intervals  in  advance  by  extending  the  <^'olumn9 
of  diflferences,  will  serve  to  enable  us  to  consider  the  neglected  terms 
partially  or  even  completely,  and  thus  derive  the  complete  perturba- 
tions for  a  ve»y  long  period.  Bui  on  account  of  the  increasing  diffi- 
culties which  present  themselves,  ari.sing  both  from  the  consideration 


510 


niLuUETICAL   ASTRONOMY. 


/ 


of  tlie  perturbations  due  to  the  action  of  the  componen*^^  Z  in  com- 
puting the  place  of  the  body,  and  from  the  magnitude  of  tiie  numeri- 
cal values  of  the  perturbations,  it  will  be  advantageous  to  determine, 
from  time  to  time,  new  osculating  elements  corresponding  to  the 
values  of  the  perturbations  for  any  particular  epoch,  and  thus  com- 
mencing the  integrals  again  with  the  value  zero,  only  the  terms  of 
the  first  order  will  at  first  be  considered,  and  the  indirect  part  of  the 
calculation  will,  on  account  of  the  smallness  of  the  terras,  be  effected 
with  great  facility.  The  mode  of  effecting  the  calculation  when  the 
higher  powers  ox'  the  mass  o  are  taken  into  account  has  already  been 
explained,  and  it  will  present  no  difficulty  beyond  that  which  is  in- 
separably connected  with  the  problem.     The  determination  of  F,  p', 

and  q'  may  be  effected  from  the  results  for  -,. ,  —-.  and  —.r  by  means 

of  the  formula;  for  integration  by  mechanical  quadrature,  as  alrca'l, 
illustrated,  or  we  may  find  F  by  a  direct  integration,  and  the  values 

of  p'  and  q'  by  means  of  the  equations  (164),  -^.-  being  found  from 

-,.j'  by  a  single  integration.     The  other  quantities  required  for  the 

com^jlcte  solution  of  the  equations  for  the  perturbations  will  be 
obtained  according  to  the  directions  which  have  been  given;  aiul  in 
the  numerical  application  of  the  formulte,  particular  attention  should 
be  given  to  the  homogeneity  of  the  several  terms,  especially  since,  for 
convenience,  we  cxpi'ess  some  of  the  quantities  in  units  of  the  seventh 
decimal  place,  and  others  in  seconds  of  arc. 

The  magnitude  of  the  perturbations  w'll  at  length  be  such  that, 
however  completely  the  terms  due  to  the  squares  and  higher  powers 
of  the  disturbing  forces  may  be  considered,  the  requirements  of  the 
numerical  process  will  render  it  necessary  to  determine  new  osculating 
elements ;  and  we  therefore  j)roceed  to  develop  the  formuloe  for  this 
purpose. 


191.  The  single  integration  of  the  vahies  of  or^-Tp  and  ft>^-^will 


dt* 


d'Sz, 
dt 

give  the  values  of  w  -tt  and  a>  —yf>  and  hence  those  of  -j.  and  -t7  '> 


dm 


which,  in  connection  with  -  -,    ,  are  required  in  the  determination  of 


dt 


dv, 


the  ..ew  system  of  osculating  elements.    Since  r"  -y~  represents  double 
the  areal  velocity  in  the  disturbed  orbit,  we  have 


in*;  Z  in  com- 
f  tiie  iiumeri- 
to  determine, 
nding  to  the 
id  thus  com- 
the  terras  of 
;t  part  of  the 
IS,  be  effected 
on  when  tlie 
already  been 
which  is  in- 
ion  of  r,  j9', 

^^   by  means 

e,  as  alread, 
d  the  vakios 

J  found  from 

ired  for  the 

ons  will  be 
^'en;  aiul  in 
ntion  should 
lly  since,  for 
tlio  seventh 


CHANGE  OF  THE  OSCULATING   ELEMENTS.  511 

dv^  __  Wp  (1  +  m) 
dt  ~  r* 

The  equation  (109)  gives 

dv^  ^_  kV^JT+mj  I         1    dm  \ 

Hence,  since  r  =  r,  (1  +  v),  we  obtain 

^=^-('  +  ^/1fJ^l  +  ^)^  (176) 

by  means  of  which  we  may  derive  ^3.     This  formula  will  furnish  at 
once  the  value  of  p,  which  appears  in  the  complete  equation  for 

-^jf.  and  also  in  the  equations  (164);  and  the  value  of  cosi  may  be 

determined  by  means  of  (165). 
In  the  disturbed  orbit  we  have 


dr       kV\  + 


m 


and  the  equations  (108)  and  (111)  give 

dr  _1c\/T+^i     .       /,    ,    1    d3M\    ,        ,  dv 

Therefore  we  obtain 


_ ,/;:  ..;„../ 1  ,  V  dm  \  ^ ^  _^  ^^  ^  ^i  ^pp^     d. 


1  r^  t,  .-in  v  =  Vp  e^  sin  v,  (  1  +   - 


^I-^i  +  m*  dt' 


Mo     dt    ' 
\vi:H    b"  .asans  of  (176),  becomes 

esiu.^e,sin.,(l+l.'^M\\i  +  ,)a^_iyZ_.4!^..         (l77^ 
The  relation  between  r  and  ?•,  gives 


P 


Po 


1  +  -  COS  V      1  +  e.j  cos  V, '     ""  ''^' 
bstituting  in  this  the  value  of  p  already  found,  we  get 


e  cos  V  —  i^l  +  e^  cos  v,) 


\    ^/'o     dt   } 


(1 +  ")'-!. 


(178) 


512  THEORETICAL  ASTRONOMY. 

Let  US  now  put 


13  = 


r,\/p 


kV\-{-m    c?<' 


(179) 


a  and  /9  being  small  quantities  of  the  order  of  the  disturbing  force, 
and  the  equations  (177)  and  (178)  become 

e  sin  V  =  e^  sin  v,  -\-  ae,,  sin  v,  -\-  /?, 
e  cos  V  =  fo  cos  V,  -\~  oCo  cos  v,  -\-  a.    - 

These  equation'^^  give,  observing  that  r,  (cos  v,  +  e^)  =j)o  ^^^  ^t 

e«i  -  v)  =  o  sini;,  —  /3  cos  v„ 

e  cos  (v,  —  v)  =  Cj  +  -^^  cos  E,-\-  p  sm  v„  '^ 

from  which  e,  v,  —  v,  and  v  may  be  found;  and  thus,  since 

X  =  ^o-\-i^,-'")>  (181) 

we  obtain  the  values  of  the  only  remaining  unknown  quantities  in 
the  second  members  of  the  equations  (164).  The  determination  of 
p'  and  q'  may  now  be  rigorously  effected,  and  the  corresjjonding 

value  of  cos  i  being  found  from  (1 65),  ~^t  and  -^  will  be  given  by 

(162).  Then,  having  found  also  1  — cos;y'  by  means  of  (166),  F  may 
be  determined  rigorously  b^  the  equation  (159),  and  not  only  the 
complete  values  of  the  perturbations  in  reference  to  all  powers  of  the 
masses,  but  also  the  corresponding  heliocentric  or  geocentric  places 
of  the  body,  may  be  found. 
If  we  put 

/  =  a  sin  V,  —  /9  cos  v„ 

3'  =  «^cosJ:,  +  /9sint;„  ^^^^^ 

and  neglect  terms  of  the  third  order,  the  equations  (180)  give 


e=e„  +  5'  +  ^, 


"» ^^  "  ^  2e; 


V,  —  v  =  —s -s, 


(183) 


in  which  s  =  206264".8.    These  equations  are  convenient  for  the 


CHANGE   OF  THE  OSCULATING  ELEMENTS. 


513 


determination  of  e  and  v,  —  v,  and  hence  X  by  means  of  (181),  when 
the  neglected  terms  arc  insensible. 
The  values  of  p,  c,  and  v  having  been  found,  we  have 


sm  ^  =  e, 


a=p  sec''  V, 


/*  = 


kVl  + 


m 


ai  (184) 

tan  \  E  =  tan  (45°  —  j  ^p)  tan  A  v,  M=  E—e sin  E, 

from  which  to  find  the  elements  f,  a,  //,  and  31.  The  mean  anomaly 
thus  found  belongs  to  the  date  t,  and  it  may  be  reduced  to  any  other 
ej)och  denoted  by  f^  by  addintr  to  it  the  quantity  fx  (<„  —  t).  When  we 
neglect  the  terms  of  the  third  order,  we  have 


V>~<Po  = 


sm  ^  —  sm  ^0 


cos  ^0  —  },  (^  —  fo)  sin  Vo 


and  if  we  substitute  for  sin  <p  —  sin  ^^  =  e  —  Cq  the  value  given  by 
the  first  of  equations  (183),  the  result  is 

2  sm  f  0  cos  f  0  —  5  sm  ^u  tan  <p^ 


from  which  we  get 


,      !f         .  ^'sin^o     , 


cos  ^j         2  cos  Vo         2  sin  ^^  cos  fp^ 


a, 


(185) 


by  means  of  which  ip  may  be  found  directly,  terms  oi'  the  third  order 
being  neglected. 

In  the  case  of  the  orbits  of  comets  for  which  c  differs  but  little 
from  unity,  instead  of  331  we  compute  by  means  of  the  formula 
(142)  the  value  of  8T,  and  since  we  have 


dt 


1    dSM 
A«„     dt 


the  equation  for  p  becomes 


and  for  a  we  have 


(186) 


(187) 


Then  c,  v,  and  g-  will  be  found  by  means  of  the  equations 

33 


514 


THEORETICAL   ASTRONOMY. 


'^ 


e  sin  (v,  —  v)  =  o  sin  v,  —  /?  cos  v„  . 
e  cos  (y,  —  v)=e^-\-  a.  (cos  v,  -j-  e,)  +  /'  sin  v„ 
P 


(188) 


and  the  time  of  perihelion  passage  will  be  derived  from  c  and  v  by 
means  of  Table  IX.  or  Table  X. 

There  remain  yet  to  be  found  the  elements  a,  Q,  and  i,  which  de- 
termine the  position  of  the  plane  of  the  disturbed  orbit  in  space. 
The  values  of  p'  and  q'  will  be  found  from  the  equations  (164),  and 
r,  whenever  it  may  be  required,  will  be  determined  as  already 
explained.     Then  we  shall  have 


sin  i  sin  (<r  —  ^^  =p\ 

sin  i  cos  (a  —  ^„)  ==  5'  -f  sin  i^, 


(189) 


from  which  to  find  i  and  a.    When  we  neglect  the  terms  of  the  third 
order,  these  equations  give 


and  hence 


,  I    pW 

sin  I  —  sm  r„  ==  q  +  -4-^1 
"      ^       sm?n 


ff=-fto  + 


—  s- 


*  =  *o4- 


pq 


q '  sm  i^ 


s, 


+  l£^^  + 


p' 


cos  Iq      '    2  cos'  ?■„ 


2  sin  if^  cos  % 


(190) 


in  which  s  =  20626  ^''.8.     The  auxiliary  spherical  triangle  which  we 
have  employed  in  the  derivation  of  the  equations  (155)  gives  directly 


cos  \  (i  +  ?■(,) 


tan^  (<r—  Q„) 


cos  ^  (t  —  io)       tan  A  (ft  —  A  +  /'o  —  ^o) 
and  since  h  —  h^  =  F,  we  have 

tan  KS^  -  S^o  -  ^0  =  S^Tfef^  tan  -A  (<r  -  ft „),       (191) 

cos  -^  [^i  -f-  ij,; 

by  means  of  which  the  value  of  ft  may  be  found.    This  equation 
gives,  when  we  neglect  terms  of  the  third  ordei'. 


ft=-fto+^  + 


fto 


sm  I 


cos  to         2  cos"  i| 


h(~^-h){<^~9,o)-      (192) 


Substituting  in  this  the  values  of  <t —  ftp  and  i  —  i^  given  by  (190), 
we  get 


ft  =  fto  + 


P 


a  — 


sm*  r 


sm  I,  cos  tj        sin'  i^  cos"  i^ 


jfpW^  +  r, 


(193) 


CHANGE   OF   THE   OSCULATING    ELEMENTS. 


515 


r  being  expressed  in  seconds  of  arc.  Finally,  for  the  longitude  of 
tlie  perihelion,  we  have 

'f  =  ;?+Q~ff,  (194) 

and  the  elements  of  the  instantaneous  orbit  are  completely  deter- 
mined. When  we  neglect ,  terms  of  the  third  order,  this  equation, 
substituting  the  values  given  by  (190)  and  (192),  becomes 


■■x  + 


tan  \  ?o 
cos  /„ 


p's  + 


tan-  \  /„  (1  +  2  cos  /„) 


9 


■J  COS"  I 


p'q's  +  /'.        (195) 


n 


It  should  also  be  observed  that  the  inclination  i  which  appears  in 
these  formulie  is  supposed  to  be  susceptible  of  any  value  from  0°  to 
180°,  and  hence  when  i  exceeds  90°  and  the  elements  are  given  in 
accordance  with  the  distinction  of  reti'ograde  motion,  they  are  to  be 
changed  to  the  general  form  by  using  180°— i  instead  of  /,  and 
2Sl  — TT  instead  of  tt. 

The  accuracy  of  the  numerical  process  may  be  checked  by  com- 
puting the  heliocentric  place  of  the  body  for  the  date  to  which  the 
ncAv  elements  belong  by  means  of  these  elements,  and  comparing  the 
results  with  those  obtained  directly  by  means  of  the  equations  (155). 
We  may  remark,  also,  that  when  the  inclination  does  not  differ  much 
from  90°,  the  reduction  of  the  longitudes  to  the  fundamental  plane 
becomes  uncertain,  and  F  may  be  veiy  large,  and  hence,  instead  of 
the  ecliptic,  the  equator  must  be  taken  as  the  fundamental  plane  to 
which  the  elements  and  the  longitudes  are  referred. 

192.  Although,  by  means  of  the  formula;  which  have  been  given, 
the  complete  perturbations  may  be  determined  for  a  very  long  period 
of  time,  using  constantly  the  same  osculating  elements,  yet,  on 
account  of  the  ease  with  which  new  elements  may  be  found  from  d3£, 

V,  dz,y  —ij-'  -.7'  and  -^^j^  and  on  account  of  the  facility  afforded  in 

the  calculation  of  the  indirect  terms  in  the  equations  for  the  differen- 
tial coefficients  so  long  as  the  values  of  the  perturbations  are  small, 
it  is  evident  that  the  most  advantageous  process  will  be  to  comj)ute 
8M,  V,  and  8z,  only  with  respect  to  the  first  power  of  the  disturbing 
force,  and  determine  new  osculating  elements  whenever  the  terms  of 
the  second  order  must  be  considered.  Then  the  integration  will 
again  commence  with  zero,  and  will  be  continued  until,  on  account 
of  the  terms  of  the  second  order,  another  change  of  the  elements  is 
required.     The  frequency  of  this  transformation  will  necessarily  de- 


516 


THEORETICAL   ASTRONOMY. 


K 


pend  oil  the  niagnitiule  of  the  disturbing  force;  and  if  the  disturbed 
body  is  so  near  tlie  disturbing  body  that  a  very  frequent  change  of 
the  elements  becomes  necessary,  it  may  be  more  convenient  either  to 
include  the  terms  of  the  second  order  directly  in  the  computation 
of  the  values  of  dJf,  v,  and  dz,,  or  to  adopt  one  of  the  other  methods 
wiiich  have  been  given  for  the  determination  of  the  perturbations  of 
a  heavenly  body.  In  the  case  of  the  asteroid  planets,  the  consider- 
ation of  the  terms  of  the  second  order  in  this  manner  will  only 
rccpiire  a  change  of  the  osculating  elements  after  an  interval  of  seve- 
ral yeai's,  and  whenever  this  transformation  shall  be  required,  the 
equations  for  <f,  i,  £1,  and  r,  in  which  the  terms  of  the  third  order 
are  neglected,  n»ay  be  employed.  It  should  be  observed,  however, 
that  the  perturbations  of  some  of  the  elements  are  much  greater  than 
the  perturbations  of  the  co-ordin.atcs,  and  hence  when  terms  depend- 
ing on  the  squares  and  higher  powers  of  the  masses  have  been 
neglected  in  the  computation  of  these  perturbations,  it  may  still  be 
necessary  to  include  the  values  of  the  terms  of  the  second  oi'der  in 
the  incomplete  equations  I'cferi'ed  to.  No  general  criterion  can  be 
given  as  to  the  time  at  which  a  change  of  the  osculating  elements 
will  be  required;  but  when,  on  account  of  the  magnitude  of  the 
values  of  dJl,  v,  and  8z„  it  appears  probable  that  the  perturbations 
of  the  second  order  ought  to  be  included  in  the  results,  by  computing 
a  single  place,  taking  into  account  the  neglected  terms,  we  may  at 
once  determine  whether  such  is  the  case  and  whether  new  elements 
are  i-equired. 


193.  We  have  already  found  the  expressions  for  the  variations  of 
Q,  and  /  due  to  the  action  of  the  disturbing  forces,  and  we  shall  now 
consider  those  for  the  variation  of  the  other  elements  of  the  orbit 
directly.  Let  x,  y,  s  bo  the  co-ordinates  of  the  body  at  any  given 
time  referred  to  any  fixed  system  of  co-ordinates.  These  will  be 
known  functions  of  the  six  elements  of  the  orbit  and  of  the  time. 
If  the  body  were  not  subject  to  the  action  of  the  disturbing  forces, 
these  six  elements  would  hi  rigorously  constant,  and  the  co-ordinates 
would  vary  only  with  the  time;  but  on  account  of  the  action  of  these 
forces  the  elements  must  be  regarded  as  continuously  varying  in  order 
that  the  relation  between  the  elements  and  the  co-ordinates  at  any 
instant  shall  be  expressed  by  equations  of  the  same  form  as  in  the 
case  of  the  undisturbed  motion.  The  co-ordinates  will,  therefore,  in 
the  disturbed  motion,  be  subject  to  two  distinct  variations:  that 
which  results  from  considering  the  time  alone  to  vary,  and  that  which 


•^;il 

■^i*!' 


VARIATION   OF   CONHTANT8.  617 

results  from  the  variation  of  the  elomcnts  themselves.  Let  these  two 
kinds  of  partial  variations  be  symbolized  respectively  by  I  -ir  \  and 
Vit\'  ^"^^  similarly  in  the  case  of  the  other  co-ordinates;  then  will 
the  total  variations  be  given  by 

'dt~\dij^  \_dtj '  'dt~\  cit  f  +  \_dt  J ' 

dt  ~\7/n+  \_dtj' 

But  if  we  differentiate  twice  in  succession  the  equations  which  ex- 
press the  values  of  x^  y,  and  z  as  functions  of  the  elements  and  of 
the  time,  regarding  both  the  elements  and  the  time  as  variable,  the 
substitution  of  tlie  results  in  the  general  equations  for  the  motion  of 
the  disturbed  body  will  furnish  three  cipiations  for  the  determination 
of  tlie  variations  of  the  elements.  There  are,  however,  six  unknown 
quantities  to  be  determinetl;  and  hence  we  may  assign  arbitrarily 
three  other  equations  of  condition.  The  supposition  which  affords 
the  required  facility  in  the  solution  of  the  problem  is  that 

[§]=«.    [!]=»'    K]=»-     ('»^) 

and  hence  that 


dx 


_(dx\  diidy\  ±__idz^\ 

~~\dtl'  dt~\dty  dt~\dtj' 


It  thus  appears  that  in  order  that  the  integrals  of  the  equations  (1) 
shall  be  of  the  same  form  as  those  of  the  equations  (3), — the  arbi- 
trary constants  of  integration  which  result  from  the  integration  of 
the  latter  being  regarded  as  variable  when  the  disturbing  forces  are 
considered, — the  first  differential  coefficients  of  the  co-ordinates  with 
respect  to  the  time  have  the  same  form  in  the  disturbed  and  undis- 

JT'    jx'  and  -T,  are  the  velocities  of  the 
dt     dt  dt 


turbed  orbits.     But  since 


disturbed  body  in  directions  parallel  to  the  co-ordinate  axes  resj)ect- 
ively,  it  follows  that  during  the  element  of  time  dt  the  velocity  of 
the  body  must  be  regarded  as  constant,  and  as  receiving  an  increment 
only  at  the  end  of  this  instant.  The  equations  (197)  show  also  that 
if  we  differentiate  any  co-ordinate,  rectangular  or  polar,  referred  to  a 


518 


THEORETICAL   ASTRONOMY. 


fixed  plane  aiul  measured  from  a  fixed  ori}j;in,  witli  respect  to  the  ele- 
meiit.s  alone  considered  as  variable,  the  first  ditferential  c«»eHicieiit 
must  he  put  equal  to  zero,  and  this  enables  us  at  once  to  effc(!t  the 
solution  of  the  problem  under  consideration.  It  is  to  be  observed, 
furthei",  that  the  functions  whose  first  diH'erential  coeflicients  with 
respect  to  the  time  when  only  the  elements  are  regarded  as  variable 
are  thus  put  equal  to  zero,  must  not  involve  directly  the  motion  of 
the  disturbed  body,  since  the  sKjcond  differential  coefficients  of  the  co- 
ordinates have  not  the  same  form  in  the  case  of  the  disturl)e<l  motion 
as  in  that  of  the  undisturbed  motion. 


iii 


194.  If  we  suppose  the  disturbing  force  to  be  resolved  into  three 
comj)onents,  namely,  E  in  the  direction  of  the  disturbed  radius- 
vector,  *S'  in  a  direction  perpendicular  to  the  i*ad ins- vector  and  in  the 
})lane  of  disturbed  orbit,  positive  in  the  direction  of  the  motion,  and 
Z  perpendicular  to  the  plane  of  the  instantaneous  orbit,  tl.c  latter 
will  only  vary  SI  and  i  and  the  longitude  of  the  perihelion  so  far  as 
it  is  affected  by  the  change  of  the  j)lace  of  the  node,  while  the  forces 
H  and  S  will  cause  the  elements  31,  ;r,  e,  and  a  to  vary  without  affect- 
ing ft  and  i. 

Let  us  now  differentiate  the  equation 


r.=P(i+.»)(?-l). 


regarding  the  elements  as  variable,  and  we  get 


rrfrl       _J_     da 


2V dV_a 

(1  +  m)  ■    rf<  ""    ' 


or 


da 
It 


2a' V 


dV 

k'  (1  +  m)  '  dt ' 


dV 


The  differential  coefficient  -r-  is  here  the  increment  of  the  accele- 

dt 

rating  force,  in  the  direction  of  the  tangent  to  the  orbit  at  the  given 

point, due  to  the  action  of  the  disturbing  force;  and  if  we  d-joignate 

the  angle  which  the  tangent  makes  with  the  prolongation  of  tiie 

radius-vector  by  (p^,  we  shall  have 

dV 

-^  =  i?  cos  v''o  +  S  sin  <Pa- 

Substituting  this  value  in  the  preceding  equation,  we  obtain 


da 

'dt 


VARIATION   OF  CONSTANTS. 

2a» 


519 


•(i?rcos^'',  +  ,SfFsiiu''J. 


P  (1  -(-  m) 
But  we  have,  according  to  tlie  equations  (50)^ 


'fi> 


Feos  4' 


'»-  Vdt)  ~ 


— -  -^ —  e  sin  V, 
Vp 


V.in,',^r{^.)=:^:lLi^-^JI^, 


in  which  v  denotes  the  true  anojnaly  in  the  instantaneous  orhitj  and 
hence  there  results 


da__ 2a" 

(^i  ~  k\/p(l-{-m) 


P  c'^ 


=.(esmvE-\-^S), 


(198) 


hy  means  of  which  the  variation  of  a  may  be  found. 
If  we  introduce  the  mean  daily  motion  fi,  we  shall  have 


and  hence 


dfi 
~dt 


if^ 


da 
a       di  ' 


-ju-  = yzj=^=:^==  (e  sm  vE  -{•  ~  S), 

dt  kVp  (1  +  w)  ^  ^  r     ^' 


(199) 
(200) 


for  the  determination  of  5//. 
The  first  of  the  equations  (97)  gives 


and  hence  we  obtain 


dt 


Hh^" 


djV'p)  _    Sr 

dt 


or 


^F 1  +  m 


dp  ^        Ipr       ^ 
dt        kVl  +  m 


(201) 


The  equation  ^  =  a  (1  —  e^)  gives 


dp        p       da       ^      de 
dt        a      dt  dt 

dp 


da 


Equating  these  values  of  -^->  and  introducing  the  value  of  — 


dt 


already  found,  we  get 

^=I7?TO(''""*+f(f-v)^)'     (202) 


/ 


620 
and  since 


THEORETICAL  AHTRONOMY. 


P  _ 


1  +  e  COS  V, 


—  =  1  —  c  cos  ^, 
a 


E  being  tlie  eccentric  unonialy  in  the  insttintuneous  orbit,  this  becomes 


lie 


dt        hV'p(\  +  m) 


{^p  sin  vR  +  p  (cos  v  -\-  cos  E)  S),      (203) 


which  will  give  the  variation  of  e.     If  we  introtluce  the  angle  of 
eccentricity  <p,  we  shall  have 


de  d<p 


p  =  a  cos'  ^, 


and  hence 

(l<p  ^  1 

di  ~  kVp  (1  +  m) 


(o  cos  <p  mi  V R -{■  a  C09  <p  (cos  v  -f  cos  E)  S),    (204) 


195.  When  we  consider  only  the  components  R  and  S  of  the  dis- 
turbing force,  the  longitude  in  the  orbit  will  be 


We  have,  therefoi*e, 


K  =  v  +  X' 


4-  =  l  +eco8(-l,  —  ;if), 


the  differentiation  of  which,  regarding  the  elements  as  variable,  gives 


-\- ernm  (X,  —  X) -^y 


or 


Th<^refore 


dp  de    .        .       ^, 

---  =  r  cos  V  -XT  +  er  sm  v  -Jr- 


dt 


dt 


dx^ 

dt' 


dy  11  7^ 

—  =  — -.  •  -  ( — »cosi;i?  +  — ^--  (2  —  cos'v  —  co8vcosi?)AS), 

dt        kl/pil+vi)     e^    ^  smv^ 

and,  I  since  p  cos  E^=^r  (cos  v  +  e),  we  have 

p{\  —  cos  V  cos  E)  =  r  sin*  v, 

so  that  the  equation  becomes 


VARIATION   OP  CONSTANTS. 


521 


^  -  -- J=^.=.  .  i-  (_j,  CO.  vJ{  +  (/>  +  r)  8iu  vS),    (205) 

IVoin  which  the  vnhio  of    .';  nmv  be  (lerived. 

(It 

If  wo  introduce  tlio  ok'incnt  (o,  or  the  niiguhir  distance  of  the  i)eri- 

helion  from  the  ascending  no(U',  it  will  be  neecssury  to  (lonsider  also 

the  contponent  Z;  and,  since  (o  -----  X  —  «t,  we  shall  have 


and  hence 


dx        'i<^  dx  .  riJi 

7lt  ~  lit  ~  Tit  ~  ^^^  ^    dt' 


dm 
It 


— ; •  — ( — pco3vE4-(p  +  r)fimvS)  —  eos  i-,—  "    ('20()) 

kVp{l-\-m)      e  >  V7  -r    /  ^^       v 


In  the  case  of  the  longitude  of  the  perihelion,  we  have 


di: 

lit 


and  therefore 
~dT 


k]/j)(i-\-m)     e 


dtu       dQ 

dt  "•"  iir 


( — p  cos  vR  -\-  (p  -\-  r)  sin  vS) 


-f2  8inM4^. 

'    dt 


(207) 


The  first  of  the  equations  (15)2  gives 
[drl         ^  .      ldM„   ,   ,,      ,,  rfM\       2r      dfi  dc       . 

in  which  J/y  denotes  the  mean  anomaly  at  the  epoch,  which  is  usually 
adopted  as  one  of  the  elements  in  the  case  of  an  elliptic  orbit.  Sub- 
stituting for  -,-  and  -^  the  values  already  found,  we  get 


dt 


kVpil-{-m) 
P 


\  (p  cot  <p  cos  V  —  2r  cos  y)  R 


or 
dt 


k\/p  (1  +  m) 
The  equation  (205)  gives 


r^  (2  —  cos'  V  —  cos  V  cos  E)  cot  vS\  —  (t  —  0-Tr» 

sin  V  ^  "  dt 


{(p  cot  <p  cos  V  —  2r  cos  y)  i2  —  (p  -\-r)  cot  <p  sin  vS) 

d/M 


-(«-<o) 


dt 


f208) 


'■■tA 


522 


THEOKETICAL  ASTRONOMY. 

1 


H'i>(l  +  m) 


(/>  4"  r)  cot  ^  pin  vS  = 


^•vXl  +  »«) 


^pcot^  eosvR 


+  COSf 


dt' 


by  means  of  which  (208)  reduces  to 


dM, 
dt 


—  COS  ^ 


dt 


2-0^^      /e-«-0-t'        (209) 


A;l/;>(l  +  w) 


(/< 


which  will  determine  the  variation  of  the  mean  anomaly  at  the 
ejioch. 

Since  the  equations  for  the  determination  of  the  place  of  the  Lody 
in  the  case  of  the  disturbed  motion  are  of  the  same  form  as  thode  for  the 
undisturbed  motion,  the  mean  anomaly  at  the  time  t  will  be  given  by 

M=^  M,  +  oM,  +  «  -  g  iih  +  <5/i), 

in  whicli  ii^  denotes  the  mean  daily  motion  at  the  instant  Iq.    There- 
fore we  shall  have 


M- 


=  J/.  +  J  '^^1  dt  +  /.„  {t  -  t,)  +  {t  -^  QJ 


fdfji 


dt, 


dt   "^   '   ''''•''      '''   '   '^''      '"V  dt 
the  integrals  being  taken  between  the  limits  t^  and  t.     The  quantity 

expresses  the  mean  anomaly  at  the  time  t  in  the  undisturbed  orbit ; 
and  if  we  designate  by  331  the  correction  to  be  applied  to  this  in 
order  to  obtain  the  mean  anomaly  in  the  dioturbed  orbit,  so  that 


SM-- 


/f* 


we  shall  have 


and  hence 


dt  ""     ^'    di    "'   '   ^''       '"V   dt 
Differentiating  this  with  respect  to  t,  we  get 

dM  _  dM,  d,.     rd,i 


dt 


dt 


dt. 


VARIATIOX   OF   COXSTAKTS. 


523 


(UL 


Substituting  in  this  the  value  of    *,  °  from  (209),  the  result  is 


dM 


dx 


-^-=:-COS^- 


2r  cos  <p 


kv'p^l-^  III 


_  /e  +  r 


dt 


dt,  (210) 


Avhioh  docs  not  involve  the  factor  t  —  /„  explicitly,  and  by  moans  of 
M-hich  the  mean  anomaly  in  the  disturbed  orbit,  at  anv  instant  /,  may 
be  found  'Erectly  from  that  for  the  same  instant  in  the  luidisturbed 
orbit. 
To  tin:!  the  variation  of  the  mean  longitude  L,  we  have 


dL       dM    ,    dr.       dx    ,    dM 


dt 


and  therefore 


do, 
dt 


(1L 
'dt 


2  sir 


dt 


,     "A     ,    O    •    2  I  -^^  2/-C0S  v* 


To  find  the  variations  of  ft  and  l,  si;ice 


)         ^ 


dfi 

dt 


dt.       (211) 


u 


V.  denoting  t\,<'  tirgument  of  the  latitude  in  the  disturbed  orbit,  M'e 
have,  according  to  the  efjuations  (169)  an  I  (170), 


dQ, 
di 

di 

"di~ 


1 


r  sni  n 


kVp  (1  +  Hi)      sm  I 
1 


Z, 


(212) 


(1  +  m) 


r  cos  w  Z. 


The  inclination  /  may  have  any  value  from  0°  to  18C°  ;  and  when- 
ever the  eletnonts  are  given  in  accordance  with  the  distinction  of  re- 
tn)gra<le  motion,  they  must  be  converted  into  those  of  the  general 
form  by  taking  180°  —  i  in  place  of  the  given  value  of /,  a;.d  2ft  — t: 
ill  place  of  the  given  value  of  ~,  before  ai)plying  the  foiuuila)  which 
involve  these  elements. 

19().  In  the  case  of  the  orbits  of  comets  in  which  the  eccentricity 
(litl'ors  but  little  from  that  of  the  parabola,  the  perturbations  of  the 
]K'rilu>lion  distance  q  and  of  the  time  of  perihelion  passage  7'  will  be 
determined  instead  of  those  of  the  elements  M  h\k\  a  or  fx. 

Tiie  equtttioa 

p  .:-.-.  5  (1  +  e) 
gives 


624 


THEORETICAL   ASTRONOMY. 


1 


dq            1 

dp 

<1 

de 

dt  ~  1  -f-  e  ' 

dt 

1+e 

dt 

and  substituting  in  this  the  value  of  -jr  already  found,  and  neglect- 
ing the  mass  of  the  comet,  which  is  always  inconsiderable,  we  get 

7 


dq 
Hi 


kVp 


1+e 


dt' 


(218) 


by  means  of  which  the  variation  of  q  may  be  found.     In  the  case  of 

de 
elliptic  motion  the  value  of  -j.  may  be  found  by  means  of  (202)  or 

(203);  but  in  the  case  of  hyperbolic  motion  the  equation  (202)  will 
be  employed.  It  should  be  observed,  also,  that  when  the  general 
formulte  for  the  ellipse  are  applied  to  the  hyperbola,  the  senil- 
cransverse  axis  a  must  be  considered  negative. 

When  the  orbit  is  a  parabola,  the  equation  (202)  becomes 


de  1 

-77-  =  — j=-  (ps>\jxvR  -{-  2p  COS^ -}iVS), 
dt        kVp  \     i-        ^      ^' 

dq 


(214) 


and  for  the  value  of  —J.-  we  have 

dt 


dq 
"dt 


JlL^S-lq^ 
kV  p  ~    at 


(215) 


It  remains  now  to  find  the  formula  for  the  variation  of  the  time  of 
perihelion  passage.     The  relation  between  T  and  M^  is  expressed  by 

360°-J/„  =  M(r-fo), 
the  differentiation  of  which  gives 


dM, 

dt 

dIL 


and,  substituting  for  -  ,.-  the  value  given  by  equation  (209),  we  get 


dT_2ar  aVp      dy  ^    1 


di 


k 


dt 


_d,^^ 
fi      dt 


Substituting  further  the  values  of  -^  and  -^  given  by  the  equations 
(205)  and  (199),  the  result  is 


VARIATION   OF  COXSTAXTS. 


525 


tlT       aR  ^^ 


p  ?jk  it  —T)      .     . 

—  cos  V —  — e  sin  v) 

e  i/p 


sin  V 


3/fc^  —  T)     p 
~~7p~ 


(216) 


'v} 


which  may  be  employed  to  determine  the  variation  of  T  wlieiievor 
the  eccentricity  is  not  very  nearly  ecjual  to  unity.  It  is  obvious, 
however,  that  when  a  is  very  large  this  e(iuation  will  not  be  con- 
venient for  numerical  calculation,  and  hence  a  further  transformation 

of  it  is  desirable.     Thus,  if  we  derive  the  expressions  for    ^    and    . 
from  tlie  equations  (24)2  ii»''^«  (-'3)2>  we  easily  obtain 

' —  .  _     -—  (J  (2r  —  —  cos  V -— — -  e  sm  v)  -\ -— — -,  cos  v, 

\-\-  e      de  e  y  j)  e  {1 -\-  ey 

2j)        dv  I  p-\-r  . 

-— : —  r  - 


■■  a  I  — 
\     e 


sin  V 


U(t  —  T)     p 


)-e-(r-b)^(i +];)--• 


1  -f  e     de         \     e  y  p 

By  means  of  these  results  the  equation  (216)  is  transformed  into 


dT_  qR  ^^  dr^  _  q 
di 


qR  .^  dr       q  a   ,    7'*^'    o    ^''    i    '// 1    ,    '"  \  •      ^       -oi-rx 

k'  ^^Te-e  '''  '^  +  k^  '^''  -Te^iV+pr'"'^-     '  -^^^ 


dT  dr 

which  may  be  used  for  the  determination  of  -.,  >  th       dues  of    .- 

and  -J-  being  found  by  means  of  the  various  formulse  develoj)(d  in 

Art.  50.  When  a  is  very  large,  its  reciprocal  denoted  by/ may  often 
he  conveniently  introduced  as  one  of  the  elements,  and,  for  the  deter- 
mination of  the  variation  of/,  we  derive  from  equation  (108) 


-^  := ^--=:  (e  sin  vR  +  ^  S). 

dt  kVp  ^  r      ^ 


(218) 


In  the  case  of  parabolic  motion  we  have  c  =  l,  and  p  =  2fj;  and 

dv  <fl! 

if  we  substitute  in  (217)  for  ~j-  and    ,    the  values  given  ])y  the  equa- 
tions (33)2  and  (30)2,  the  result  is 

(IT  0*  /  R 

1-T-; -TT-     v.-  (-  1  +  '^  tan'  \  V  +  tan*  A  v  +  J,  tan"  A  lO 
1  -f-  tau^  ^v\  k'  ^ 

+  -^(4tanAt;-|tan»^v))-  (219) 


dt 


626 


THEORETICAL    ASTKONOMY. 


197.  Instciid  of  tlie  elements  usually  employed,  it  may  be  desirable, 
in  rare  and  .spcoial  cases,  to  introduce  otber  combinations  of  the  ele- 
ments or  constants  which  determine  the  circumstances  of  the  undis- 
turbed motion,  and  the  relation  between  the  new  elements  adopted 
and  those  for  which  the  expressions  for  the  diff'ex'ential  cocHica'iit.s 
have  been  j^iven,  will  furnish  immediately  the  necessary  fornuilii'. 
In  the  case  of  the  periodic  comets,  it  will  o"l:en  be  desired  to  dotcr- 
mine  the  alteration  of  the  periodic  time  arising  from  the  action  of  tlio 
disturbing  planets.  Let  us,  therefore,  suppose  that  a  comet  has  been 
identified  at  two  successive  returns  to  the  perihelion,  and  let  r  doiidte 
the  elapsed  interval.  The  observations  at  each  appoaraiice  of  the 
comet,  however  extend(>d  tliey  may  be,  will  not  indicate  with  certainty 
the  semi-transverse  axis  <»f  the  orbit,  and  hence  the  periodic  time. 
But  when  r  is  known,  by  eliminating  the  effect  of  the  disturbing 
forces,  we  may  determine  with  accuracy  the  value  of  the  semi-trans- 
verse axis  a  at  each  epoch,  and,  from  this  and  the  observed  places, 
the  other  elements  of  the  orbit  according  to  the  process  already 
explained. 

Let  /^u  be  the  mean  daily  motion  at  the  first  epoch,  and  we  shall 
have 


f^o'^ 


+/^ 


dt 


dt  =  2ff, 


in  which  -  denotes  the  semi-circumference  of  a  circle  whose  radius  is 
unity.     Hence  we  obtain. 


/ 


dM 


2.-|-^^-rf« 


(220) 


by  means  of  which  to  determine  n^^.  Then,  to  find  the  mean  daily 
motion  ji  at  the  instant  of  the  second  return  to  the  perihelion,  we 
have 

^-''^+St'''  (221) 


the  integral  being  taken  between  the  limits  0  and  r.  The  provisional 
value  of  the  mean  motion  as  given  by  the  observed  interval  r  will  be 
sufficiently  accurate  for  the  calculation  of  the  variations  of  M  and  ji 
during  this  interval.  The  semi-transverse  axis  will  now  be  derived 
by  means  of  the  formula 


a=\ 


\1. 


VARIATION   OF  CONSTANTS. 


627 


from  the  values  of  fx  for  the  t>vo  epochs.  Let  r'  denote  tlie  interval 
which  must  elapse  before  the  next  succeeding  perihelion  passage  of 
tlie  comet,  and  we  have 


and  consequently 


lz  =  „z'JrS- 


dM 
dt 


dt, 


^dM 


1^  = 


U-p-^^M 


(222) 


se  raLiius  is 


the  integral  being  taken  between  the  limits  1  =  0,  corresponding  to 
the  beginning  of  the  interval,  and  t  =  r'.     We  have,  therefore, 


8t 


-\P>- 


(223) 


for  the  change  of  the  periodic  t5.ne  due  to  the  action  of  the  disturb- 
ing forces. 

198.  The  ct^lculation  of  the  values  of  the  components  R,  S,  and  Z 
of  the  disturbing  force  will  be  effected  by  means  of  the  formulte 
given  in  Art.  182.  It  will  be  observed,  however,  that  not  only  tliese 
components  of  the  disturbing  force,  but  also  their  coefficients  in  the 
expressions  for  the  differential  coefficients,  involve  the  variable  ele- 
ments, and  hence  the  perturbations  which  are  sought.  But  if  we 
consider  only  the  pertui'bations  of  the  first  order,  the  fundamental 
osculating  elements  may  be  employed  in  place  of  the  actual  variable 
elements,  and  whenever  the  perturbations  of  the  second  order  have  a 
sensible  influence,  the  elements  must  be  corrected  for  the  terms  of  the 
first  order  already  obtained.  Then,  commencing  the  integration  anew 
at  the  instant  to  which  the  corrected  elements  belong,  the  calculation 
may  be  continued  until  another  change  of  the  cL  ncnts  becomes 
necessary.  The  several  quantities  required  in  the  computation  of  the 
forces  may  also  be  corrected  from  time  to  time  as  the  elements  are 
changed. 

The  frequency  with  which  the  elements  must  be  changed  in  order 
to  include  in  the  results  all  the  terms  which  have  a  sensible  influence 
iu  the  determination  of  the  place  of  the  disturbed  body,  will  depend 
entirely  on  the  circumstances  of  each  particular  case.  In  the  case  of 
the  asteroid  planets  this  change  will  generally  be  required  only  after 
an  interval  of  about  a  year;  but  when  ^he  planet  apjiroaches  very 
near  to  Jupiter,  the  interval  may  ne -essarily  be  much  shorter.     The 


n 


'ii 


528 


THEORETICAL   ASTRONOMY. 


magnitude  of  the  resulting  values  of  the  perturbations  will  suggest 
the  necessity  of  correcting  the  elements  whenever  it  exists;  and  if 
we  apply  the  [)roper  corrections  and  commence  anew  the  integration 
for  one  or  more  intervals  preceding  the  last  date  for  which  the  per- 
turbations of  the  first  order  have  been  found,  it  will  appear  at  once, 
by  a  comparison  of  the  results,  whether  the  elements  have  too  long 
been  regarded  as  constant. 

The  intervals  atwi'i!  the  differential  coefficients  must  be  com- 
puted directly,  will  also  depend  on  the  relation  of  the  motion  of  the 
disturbing  body  to  that  of  the  disturbed  body;  and  although  the  in- 
terval may  be  greater  than  in  the  case  of  the  variations  of  the  co- 
ordinates which  require  an  indirect  calculation,  still  it  must  not  be  so 
large  that  the  places  of  both  the  disturbing  and  the  disturbed  body,  as 
well  as  the  values  of  the  several  functions  involved,  cannot  be  inter- 
polated with  the  requisite  accuracy  for  all  intermediate  dates.  In  tlie 
case  of  the  asteroid  planets  a  uniform  interval  of  about  forty  days  will 
generally  be  preferred;  but  in  the  case  of  the  comets,  which  rapidly 
approach  the  disturbing  body  and  then  again  rapidly  recede  from  it, 
the  magnitude  of  the  proper  interval  for  quadrature  will  be  very 
diiferent  at  different  times,  and  the  necessity  of  shortening  the  inter- 
val, or  the  admissibility  of  extending  it,  will  be  indicated,  as  the 
numerical  calculation  progresses,  by  the  manner  in  which  the  several 
functions  change  value. 

If  we  compute  the  forces  for  several  disturbing  bodies  by  using 
I'R,  1\S,  and  I'Z  in  the  formulae  in  place  of  R,  S,  and  Z,  respect- 
ively, the  total  perturbations  due  to  the  combined  action  of  all  of 
these  bodies  may  be  computed  at  once.  But,  although  the  numerical 
process  is  thus  somewhat  abbreviated,  yet,  if  the  adopted  values  of 
the  masses  of  some  of  the  disturbing  bodies  are  uncertain,  and  it  is 
desired  subsequently  to  correct  the  results  by  means  of  corrected 
values  of  these  masses,  it  will  be  better  to  compute  the  perturbations 
due  to  each  disturbing  body  separately,  and,  since  a  large  part  of  the 
numerical  process  remains  unchanged,  the  additional  labor  will  not 
be  very  considerable,  especially  when,  for  some  of  the  disturbing 
bodies,  the  interval  of  quadrature  may  be  extended.  The  successive 
correction  of  the  elements  in  order  to  include  in  the  results  the  per- 
turbations due  to  the  higher  powers  of  the  masses,  must,  however, 
involve  the  perturbations  due  to  all  the  disturbing  bodies  considered. 

The  differential  coefficients  should  be  multiplied  by  the  interval  w, 
so  that  the  formulae  of  integration,  omitting  this  factor,  will  furnish 
directly  the  required  integrals;  and  whenever  a  change  of  the  inter- 


NUMERICAL   EXAMPLE. 


529 


val  h  introduced,  thf  proper  caution  must  be  observed  in  regard  to 
the  process  of  integration.  The  quantity  s  =  200264". 8  should  be 
uitroduced  into  the  fornmlie  in  such  a  manner  that  the  variations  of 
the  elements  which  are  expressed  in  angular  measure  will  be  obtiiined 
directly  in  seconds  of  ai*c ;  and  the  variations  of  the  otlier  elements 
will  be  conveniently  determined  in  units  of  the  nth  decimal  place. 
It  should  be  observed,  also,  that  if  the  constants  of  integration  are 
put  equal  to  zero  at  the  beginning  of  the  integration,  the  integrals 
obtained  will  be  the  required  perturbations  of  the  elements. 

199.  Example. — We  shall  now  illustrate  the  calculation  of  the 
perturbations  of  the  elements  by  a  numerical  example,  nnd  for  this 
purpose  we  shall  take  that  which  has  already  been  solved  by  the 
other  methods  which  have  been  given.  From  1864  Jan.  1.0  to  1865 
Jan.  15.0  the  perturbations  of  the  second  order  are  insensible,  and 
hence  during  the  entire  period  it  will  be  sufficient  to  use  tiie  \  olues 
of  r,  V,  and  E  given  by  the  osculating  elements  for  1864  Jan.  1.0. 

The  calculation  of  the  forces  it,  S,  and  Z  is  effected  precisely  as 
already  illustrated  in  Art.  189,  and  from  the  results  there  given  we 
obtain  the  following  values  of  the  forces,  with  which  we  vrite  also 
the  values  of  Eg: — 


Berlin  Mean  Time. 

40R 

40^ 

AOZ 

Eo 

1863  Dec. 

12.0, 

+  0".0365 

+  0".0019 

+  0".00002 

355° 

26'  8' 

.2 

1864  Jan. 

21.0, 

0  .0356 

—  0  .0086 

0  .00025 

8 

14  57 

.8 

March 

1.0, 

0  .0315 

0  .0182 

0  .00047 

20 

57  55 

.1 

April 

10.0, 

0  .0250 

0  .0259 

0  .00068 

33 

26  47 

.6 

May 

20.0, 

0  .0169 

0  .0314 

0  .00087 

45 

35  25 

.3 

June 

29.0, 

+  0  .0079 

0  .0343 

0  .00101 

•''7 

20  3 

.8 

Aug. 

8.0, 

—  0  .0011 

0  .0349 

0  .00112 

68 

39  14 

.6 

Sept. 

17.0, 

0  .0099 

0  .0333 

0  .00117 

79 

33  13 

.1 

Oct. 

27.0, 

0  .0179 

0  .0301 

0  .00116 

90 

3  23 

.2 

Dec. 

6.0, 

0  .0252 

0  .3253 

0  .00108 

100 

11  49 

.1 

1865  Jan. 

15.0, 

—  0  .0317 

—  0  .0193 

+  0  .00090 

110 

0  54 

.3 

We  compute  the  values  of  the  required  differential  coefficients  by 
means  of  the  equations 


dsa 


r  sm  u 


Z, 


dSi 
It 


r  cos  «  Z, 


dSK 
dt 


dt         kVp        si^*"  "'  dt        jci/p 

1     /  _  ^cos^  ^      (p+^^  A        ^.^,  ^.d^^ 
\/p  \        s\n  <p  sm  v"  /  ~     dt 


hV 


34 


530 


THEORETICAL  ASTRONOMY. 


--—  =  — -^  (a  COS  y9mvR-\-acos<p  (cos  v  -\-  cos  E)  S), 
at         kv  p 

ddfi  _ 
"dt" 


•y=  •  —  (sin  ^  sm  vR  +  -  S), 


dm 

'dt 


kVjy 


77,— I    ^. 2r]R  —  ^^   '.  ^ /S|cos^+  1-77- rf<; 

&l/p  \  \  sin  9?  /  sin  ^  /        "^    '  c/    (/<       ' 


and  the  results  are  the  following: — 


Sate. 

„£.? 

»^ 

«1r 

„« 

1600  ''*'' 
dt 

-f^r" 

40  ^'1' 

fit 

1803  Doc. 

12.0, 

—  0".004 

—  O".0Ol 

— 16".730 

+  0".022 

—  0".0790 

+  0".027 

+  ll".092 

1804  .Tim. 

21.0, 

0  .108 

0  .017 

17  .255 

—  0  .992 

+  0  .4524 

0  .102 

11  .864 

Miirch   1.0, 

0  .SO-2 

0  .026 

l5  .578 

1  .810 

0  .9396 

0  .863 

15  .381 

April 

10.0, 

0  .555 

0  .028 

22  .986 

2  .294 

1  .3321 

2  .008 

20  .746 

Mny 

20.0, 

0  .822 

0  .022 

26  .572 

2  .418 

1  .6169 

3  .492 

26  .893 

Juno 

29.0, 

1  .037 

—  0  .007 

29  .271 

2  .228 

1  .7750 

-)  .198 

32  .617 

Aug. 

8.0, 

1  .189 

+  0  .012 

30  .698 

1  .829 

1  .8196 

V  .004 

37  .293 

Sept. 

17.0, 

1  .233 

0  .033 

30  .500 

I  .406 

1  .7591 

8  .801 

40  .445 

Oct. 

27.0, 

1  .169 

0   052 

28  .953 

1  .055 

1  .6206 

10  .498 

42  .144 

Dec. 

6.0, 

1  .004 

0  .005 

26  .498 

0  .902 

1  .4074 

12  .017 

42  .741 

1865  Jan. 

15.0, 

—  0  .742 

+  0  .oor 

—  23  .336 

—  1  .004 

+ 1  .1388 

+  13  .292 

+  42  .3a 

The  values  thus  obtained  give,  by  means  of  the  formulae  for  integra- 
tion by  mechanical  quadrature,  the  following  perturbations  of  the 
elements : — 


Berlin  Mean  Time. 

m 

Si 

Sit 

«« 

V 

J.W 

1863  Dec.     12.0, 

+  O^'.Ol 

—  O'^OO 

+  8'^43 

+  0'M2 

+  0'^0007 

—  5'M8 

1864  Jan.     21.0, 

-0   .04 

0   .01 

—  8    .49 

—  0   .38 

0 

.0040 

+  5  .72 

March    1.0, 

0   .24 

0   .03 

26   .78 

1   .80 

0 

.0216 

19  .15 

April   10.0, 

0   .66 

0    .06 

48   .01 

3   .88 

0 

.0502 

37  .11 

May     20.0, 

1    .35 

0   .08 

72   .82 

6   .27 

0 

.0875 

60  .91 

June    29.0, 

2   .28 

0   .10 

100   .83 

8   .61 

0 

.1299 

90  .73 

Aug.      8.0, 

3   .40 

0   .09 

130   .93 

10   .65 

0 

.1751 

125  .79 

Sept.    17.0, 

4   .63 

0   .07 

161    .66 

12   .26 

0 

.2200 

164  .79 

Oct.      27.0, 

5   .84 

—  0   .03 

191    .48 

13   .48 

0 

.2624 

206  ,19 

Dec.       6.0, 

6   .93 

+  0   .03 

219    .27 

14   .44 

0 

.3004 

248  .72 

1865  Jan.     15.0, 

—  7    .81 

+  0   .10 

-244   .24 

-15   .37 

+  0 

.3323 

+  291   .33 

Applying  the  variations  of  the  elements  thus  obtained  to  the  oscu- 
lating elements  for  1864  Jan.  1.0,  as  given  in  Art.  166,  the  o,sculating 
elements  for  the  instant  1865  Jan.  15.0  are  found  to  be  the  following: — 

Epoch  =  1865  Jan.  15.0  Berlin  mean  time. 


JI/=   99°34'48".81 


r  : 

=    44    13     7 

.93) 

SI 

=  206   38  57 

.88  y 

i 

=     4   36  52 

.21  i 

9  -- 

=   11    15  35 

.65 

log  a: 

=  0.3880283 

/*  = 

=  928".8897. 

(  Ecliptic  and  Mean 
Equinox  1860.0. 


NUMERICAL   EXAMPLE, 


531 


't,.„« 

027       +  U" 

092 

102           U 

861 

S63           15 

381 

IX)8           20 

746 

492           26 

893 

198           32 

.617 

004           3" 

.293 

SOI           40 

.445 

498           42 

.144 

017           42 

.741 

292       +  42 

.323 

or  integra- 

ons  of  the 

jj/ 

)7      —  5".48 

0      +5 

.72 

6         19 

.15 

)2        37 

.11 

5         60 

.91 

19         90 

.73 

il       125 

.79 

)0       164 

.79 

!4       206 

.19 

14       248 

.72 

13  +  291 

.33 

0  the  oscu- 

>  osculat 

hlg 

llowiug 

: — 

In  order  to  compare  the  results  thus  derived  with  the  perturbations 
computed  by  the  other  methods  whicli  have  been  given,  let  us  com- 
pute the  heliocentric  longitude  and  latitude,  in  the  case  of  the  dis- 
turbed orbit,  for  the  date  1865  Jan.  15.0,  Berlin  mean  time.  Thus, 
by  means  of  the  new  elements,  we  find 


M=   99°  34' 48".81, 
logr=   0.4162182, 
Z=:164°37'59".04, 


E-=  110°  5'14".15, 
V  =--  120  19  18  .01, 
6  =  —  3      5  32  .54, 


agreeing  completely  Avith  the  results  already  obtained  by  the  other 
methods.  The  heliocentric  place  thus  found  is  referred  to  the  ecliptic 
and  mean  equinox  of  1860.0,  to  which  the  elements  rr,  Q,  and  i  are 
referred ;  and  it  may  be  reduced  to  any  other  ecliptic  and  equinox  by 
means  of  the  usual  formulae.  Throughout  the  calculation  of  the  per- 
turbations it  will  be  convenient  to  adopt  a  fixed  equinox  and  ecliptic, 
the  results  being  subsequently  reduced  by  the  application  of  the  cor- 
rections for  precession  and  nutation. 

In  the  determination  of  d3I,  if  we  denote  by  JM  the  value  which 

is  obtained  when  we  neglect  the  last  term  of  the  equation  for  ~ir->  we 
shall  have 


«53/=-  dM 


+J[j(^< 


which  form  is  equally  convenient  in  the  numerical  calculation.   Thus, 
for  1865  Jan.  15.0,  we  find 

AM==  +  234".74, 

and  from  the  several  values  of  1600—^  we  obtain,  for  the  same  date, 
by  means  of  the  formula  for  double  integration. 


//**  = +  56-.69, 


dt 
Hence  we  derive 

dM=-\-  234".74  +  56".59  =  +  291".33, 

agreeing  with  the  result  ah'eady  obtained. 

If  we  compute  the  variation  of  the  mean  anomaly  at  the  epoch,  by 
means  of  equation  (209),  we  find,  in  the  case  under  consideration. 


dMa  =  +  165".29, 


532 


tiip:oretical  astronomy. 


I  / 


ami  since  the  place  of  the  body  in  the  case  of  the  instantaneous  orbit 
is  to  be  computed  precisely  as  if  the  planet  had  been  moving  con- 
stantly in  that  orbit,  wc  have,  for  18G5  Jan.  15.0, 


and  hence 


(<-goV  =  4-126".27,   ' 
dM=  'Uf,  -\-(t  —  g  ¥  =  +  291".56. 


The  error  of  this  result  is  —  0".23,  and  arises  chioHy  from  the  in- 
crease of  the  accidental  and  unavoidable  errors  of  the  numerical  cal- 
culation by  the  factor  t  —  /„,  which  appears  in  the  last  term  of  the 
equation  (209).  Hence  it  is  evident  that  it  will  always  be  preferable 
to  compute  the  variation  of  the  mean  anomaly  directly;  and  if  the 
variation  of  the  mean  anomaly  at  a  given  epoch  be  required,  it  may 
easily  be  found  from  831  by  means  of  the  equation 

If  the  osculating  elements  of  one  of  the  asteroid  planets  are  thus 
determined  for  the  date  of  the  opposition  of  the  planet,  they  will 
suffice,  without  further  change,  to  compute  an  ephcmcris  for  the  brief 
period  included  by  the  observations  in  the  vicinity  of  the  opposition, 
unless  the  disturbed  planet  shall  be  very  near  to  Jupiter,  in  which 
case  the  perturbations  during  the  period  included  by  the  ephcmeris 
may  become  sensible.  The  variation  of  the  geocentric  place  of  the 
disturbed  body  arising  from  the  action  of  the  disturbing  forces,  may 
be  obtained  by  substituting  the  corresponding  variations  of  the  ele- 
ments in  the  differential  formula)  as  dei'ived  from  the  equation  (1)2, 
whenever  the  terms  of  the  second  order  may  be  neglected.  It  should 
be  observed,  however,  that  if  we  substitute  the  value  of  oil/ directly 
in  the  equations  for  the  variations  of  the  geocentric  co-ordinates,  the 
coefficient  of  d/i  must  be  that  which  depends  solely  on  the  variation 
of  the  semi-transverse  axis.  But  when  the  coefficient  of  d/i  has  been 
computed  so  as  to  involve  the  effect  of  this  quantity  during  the  in 
terval  t 
tuted  in  the  equations 


t^,  the  value  of  oMg  must  be  found  from  oM  and  substi- 


200.  It  will  be  observed  that,  on  account  of  the  divisor  e  in  the 

expressions  for  -^.  -tt'  and  -^->  theseelements  will  be  subject  to  large 

perturbations  whenever  e  is  very  small,  although  the  absolute  effect 
on  the  heliocentric  place  of  the  disturbed  body  may  be  small;  and  on 


VARIATION  OP  CONSTANTS. 


633 


account  of  the  divisor  sin  t  in  the  expression  for     ,,    the  variation 

'  (It 

of  SI  will  be  large  whenever  i  is  very  small.     To  avoid  the  difHciil- 

tios  thus  encountered,  new  elements  nuist  be  introdueetl.     Thus,  in 

the  ease  of  SI ,  let  us  put 

a"  ==  sin  i  8iu  Si,  /5"  =  sin  i  cos  SI ;  (224) 

then  we  shall  have 

da"        .     _        .di  .  dSl 

-  .,  -  =  sm  SI  cos  t-jj-  -\-  sm  i  cos  SI    ,.  > 

dr  _         .  (//         .     .  .    ^dSi 

-J.-  =  cos  SI  COS  I  ,-  —  sui  I  sni  SI    ,,  • 

at  at  at 

Introducing  the  ^'alucs  of  -r-  and  given  by  the  equations  (212), 

and  introducing  further  the  auxiliary  constants  a,  b,  A,  and  B  com- 
puted by  me.'iiis  of  the  fornuila)  (94)i  with  respect  to  the  fundamental 
plane  to  which  Si  and  i  are  referred,  we  obtain 


da." 


dt 

dt 


kVp  (1  +  m) 
1 


rZ  sin  a  cos  (A  -\-  u), 


(225) 


kV])  (1  +  m) 


— ^TTT^  rZ  sin  b  cos  (B  +  u), 


by  means  of  which  the  variations  of  «"  and  ft"  may  be  found.  If 
the  integrals  are  put  equal  to  zero  at  the  beginning  of  the  integration, 
the  values  of  da"  and  dft"  will  be  obtained,  so  that  we  shall  have 

sin  i  sin  Ji  =  sin  ig  sin  Slo-\'  ^*"> 
sin  i  cos  SI  =  sin  tj  cos  S^o  -|-  Sfi", 


or 


sin  i  sin  (  Si  —  SJo)  =  cos  Sio  '^«"  —  sin  Slo  'W, 

sin  i  cos  (Si  —  J^^)  =  sin  i^  +  sin  J^o  da"  -f  cos  Slo^f^"> 

by  means  of  which  i  and  Si  —  Sio  '"^y  be  found. 
In  the  case  of  ;f,  let  us  put 


and  we  have 


e  sin  X, 


dr," 


de 


f "  =  e  cos  x, 


dx 


(226) 


(227) 


^dr='''''^-dt-^''''^-'dr 


d:"^ 

dt 


de 


dx 


cosx-^-esmx^^ 


534 


THEOUETICAL   ASTUONOMY. 


Substituting  for        and    f  tho  values  given  by  the  equations  (203) 
and  (205),  and  reducing,  wc  obtain 


df^  1 


dt 


I  —  p  cos  (y  -\-  x)  R  •\-  \(p  -\-  '■)  sin  {v  -\-  x) 
-l-cramxl^j, 

'  1  / 

"  =  ,./   ,,    ,      Ap  sin{v  +  /)]i-\-  Kj)  +  r)  cos  (v  +  jr) 

4-  e7'co8/(^S'|, 


(228) 


by  moans  of  wliich  tl>o  values  of  3r/'  and  3!^"  may  be  found.     Then 
■sve  shall  have 

e  sin  x=^e„  sin  r„  +  ,^7)", 
1  e  cos  /  =  fo  *^o^  ''o  "f  ''»"> 


or 


e  sin  f  ;^  —  rr^)  ==  cos  r„  ^jj"  —  sin  rr^  rJC", 

e  cos  ix  —  '^o)  =  ^0  +  s'"  ^0  ^v"  -\-  cos  tTq  K", 


(229) 


from  which  to  find  c  and  ;f.    If,  in  order  to  find  the  variation  of  z,  we 

write  t:  instead  of  ;f  in  these  formula',  the  termrj  +  2ccos7rsin'^  i*  7; 

and  —  2e  sin  TT  sin'Hi -,7-  must  be  added  to  the  second  members  of 
-^    dt 

(228),  respectively. 


201.  By  means  of  the  four  methods  which  we  have  develoiicd  and 
illustrated,  the  special  perturbations  of  a  heavenly  body  may  be  de- 
termined with  entire  accuracy,  and  the  choice  of  the  j)articular  method 
will  depend  on  the  circumstances  of  the  case.  By  computing  the 
perturbations  of  the  elements,  correcting  these  elements  as  often  as 
may  be  required,  the  terms  depending  on  the  higher  powers  of  the 
masses  may  be  included,  and  no  indirect  calculation  becomes  necessary. 
The  frequent  correction  of  the  elements  will  also  render  insensible 
the  effect  of  whatever  uncertainty  remains  in  regard  to  their  true 
values.  But,  since  the  perturbations  of  the  elements  are  in  general 
much  greater  than  those  of  the  co-ordinates,  the  effect  of  the  terms 
of  tlie  second  order  will  be  much  greater  upon  the  values  of  the  ele- 
ments than  upon  those  of  the  co-ordinates.  Hence,  the  frequency 
with  which  a  change  of  the  elements  will  be  required  will  fully  com- 
pensate the  labor  of  the  indirect  part  of  the  calculation  in  the  case 
of  the  perturbations  of  the  co-ordinates. 


VARIATIOX  OF  COXHTANTS. 


530 


The  (Ictermlimtion  of  tlu;  poi'tiirl)ation8  of  the  pohir  eo-onliiiatort 
r,  w,  and  ;,  ami  tluit  of  the  |)orturl)iitioii.s  d.]f,  u,  and  dz„  arc  t'tll-ctt'd 
with  ahiiost  etjual  facility,  espcoially  when  tin'  eH'cct  of  the  dishirl)- 
in}f  forces  is  to  he  determined  ibr  a  long  interval  of  lime.  If  the 
perturhatioiiH  are  recjuired  oidy  for  a  hrief  jteriotl,  it  will  be  preicr- 
ahle  to  determine  r?J/,  u,  and  (h,  rather  than  die,  rr,  and  2,  since  tl»e 
indirect  part  of  the  calculation  will  thus  bo  effected  with  less  repe- 
tition. In  both  of  these  awes  the  values  of  the  perturbations  are 
fienerally  smaller  than  in  the  case  of  the  rectanjjjniar  co-ordinates,  and 
hence  they  are  less  affected  by  terms  of  the  second  order;  but  on 
account  of  the  simplicity  of  the  fornudie,  even  when  we  include  the 
terms  depending  on  the  higher  powers  of  the  masses,  so  long  as  the 
nuigniiude  of  the  values  of  o.r,  rii/,  and  dz  is  not  so  large  as  to 
render  troublesome  the  indirect  part  of  the  calculation,  the  method 
of  the  variation  of  rectangular  co-ordinates  may  be  advantageously 
employed  when  the  perturbations  are  to  be  determined  for  a  long 
period. 

By  whatever  method  the  perturbations  are  determined,  if  the  fun- 
damentsU  osculating  elements  are  correct,  the  final  elements  of  the 
instantn  'cous  orbit  will  be  the  same.  But,  since  the  effect  of  the 
error  of  *he  elements  will  differ  in  degree  in  the  different  methods 
of  treating  the  problem,  if  these  elements  are  aflected  with  small 
errors,  the  agrcoiuent  of  the  final  os(!ulating  elements  obtained  by  the 
different  iiiethods,  in  connection  with  the  corrections  derived  by  the 
conij)arison  of  observations,  may  not  be  complete. 

When  the  disturbed  body  approaches  very  near  to  a  disturbing 
planet,  the  magnitude  of  the  perturbations  will  be  such  as  to  enable 
us  by  means  of  accurate  observations  to  correct  the  adopted  value  of 
the  disturbing  mass.  In  this  case  the  perturbations,  computed  by 
means  of  either  of  the  methods  applicable,  must  be  converted  into 
the  corresponding  perturbations  of  the  geocentric  spherical  co-ordi- 
nates. Let  the  variation  of  either  of  the  geocenh'ie  co-ordinates 
arising  from  the  action  of  the  disturbing  planet  be  denoted  by  (W; 
then,  if  we  suppose  the  correct  value  of  the  disturbing  mass  to  be 
1  +  n  times  the  assumed  value  used  in  con)i)uting  (W,  the  correspond- 
ing variation  of  the  geocentric  spherical  co-ordinate  will  be 

(1  +  n)  do. 

The  value  dd  may  be  included  in  the  determination  of  the  difference 
between  computation  and  observation  in  the  formation  of  the  equa- 
tions of  condition  for  finding  the  corrections  to  be  applied  to  the  ele- 


536 


TIIEOKETICAI.   ASTIIOXOMY. 


/ 


rncnts;  and,  finally,  the  term  nod  may  be  added  to  each  oi  the  equa- 
tions of  condition,  so  that  we  thu.s  introduce  a  new  unknown  quantity 
71.  The  solution  of  ali  the  equations  thus  formed,  by  the  method  of 
least  squai'j,  will  then  furnish  the  most  probable  values  of  the  cor- 
rections to  be  applied  to  tlie  adopted  elements,  and  also  the  value  of 
11,  by  means  of  which  a  corrected  value  of  the  mass  of  tlie  disturbing 
body  will  be  obtained. 

202.  If  the  determination  of  the  perturbations  of  a  heavenly  body 
required  that  all  the  disturbing  bodies  in  the  system  should  be  con- 
stantly considered,  the  labor  would  be  very  great.  But,  fortimatoly, 
it  so  happens  that  the  nidsses  of  many  of  the  planets  arc  so  small  in 
comparison  with  that  of  the  sun,  that  the  si>hcre  of  their  disturbing 
influence  is  very  much  restricted.  Thus,  in  the  determination  of  the 
perturbations  of  the  asteroid  planets,  only  the  action  of  Max's,  Jupi- 
ter, and  Saturn  need  be  considered;  and  of  these  disturbing  planets 
Ju]>iter  exerts  the  principal  influence.  It  is  true,  however,  that,  on 
account  of  the  elongated  form  of  the  orbits  of  the  periodic  comets, 
tliey  may  at  different  times  be  sensibly  disturbed  by  each  of  the 
planets  of  the  system.  But  since  in  the  remote  parts  of  their  orbits 
they  are  very  distant  from  many  of  the  disturbing  planets,  the  deter- 
mination of  their  perturbations  will  th<m  be  much  facilitated  by  con- 
sidering them  as  revolving  around  the  common  centre  of  gravity  of 
the  sun  and  disturbing  planet.  When  the  motion  is  referred  to  the 
centre  of  the  sun,  the  disturl)ing  force  is  the  difference  of  the  direct 
action  of  the  disturbing  body  upon  the  disturbed  l)ody  and  ui)on  the 
sun ;  and  in  the  case  of  those  disturbing  planets  whose  periodic  time 
is  short,  the  term  which  expresses  the  action  upon  the  sun  Avill  change 
value  so  rapidly  that  it  will  l)e  necessary  to  adopt  small  intervals  in 
the  direct  numerical  calculation.  But  when  we  refer  the  motion  to 
the  centre  of  gravity  of  the  system,  which  does  not  receive  any 
motion  in  virtue  of  the  mutual  attractions  of  the  bodies  which  com- 
pose the-system,  that  part  of  the  disturbing  force  which  expresses  the 
action  of  the  disturlnng  planet  upon  the  sun  will  disappear,  and  tlio 
magnitude  of  the  disturbing  force  will  be  less  th.an  that  of  the  ioroe 
which  disturbs  the  motion  of  the  comet  relative  to  the  sun,  so  that 
the  intervals  for  quadrature  may  be  greatly  extended.  It  will  bo 
observed,  further,  that,  if  the  distance  of  the  comet  from  tlie  sun  i^ 
far  greater  than  the  distance  of  the  disturl)ing  b.^'|y,  the  direct  action 
of  the  planet  upon  the  comet  becomes  so  small  that  its  effect  upon  the 
motion  will  be  quite  insignificant.     In  this  ease  the  motion  of  the 


PERTURBATIONS  OF  COMETS. 


537 


oomet  will  be  sensibly  the  same  as  the  pure  elliptic  motion  around 
the  common  centre  of  gravity  of  the  sun  and  disturbing  planet. 

In  order  to  exhibit  these  principles  more  clearly,  let  us  denote  by 
s,  jy,  C,  the  co-ordinates  of  the  sun  referred  to  the  centre  of  gravity 
of  the  system;  by  .r,,,  i/^,  z^^,  the  co-ordinates  of  the  comet;  and  by 
■xj,  yj,  Zq',  the  co-ordinates  of  the  disturbing  jjlanet  referred  to  the 
same  origin.  Let  x,  y,  z  be  the  co-ordinates  of  the  comet,  and 
x',  y',  z'  those  of  the  planet  referred  to  the  centre  of  the  sun;  then 
we  shall  have 


l  +  .r, 


ar„ 


■T  =  —  tn  x^ , 


Vo 


V  +  y, 


»^'ydi 


and  hence 

X  —Xg  -{■  Hi'.r;, 
x'  =  xl  -f-  «i'a-(,', 

From  these  we  derive 


2/  =  y„  H-  m'yl, 
y'  =  2/0'  +  »''yo'. 

r  ~  »"o  +  wi  ^0  • 


m'x' 


m 


Zfy '■     -,      -j-     Z, 

C  ^  —  m'z^, 


3  —  2o  +  "''-'o'» 
z'  —  2„'  -f  m'z^, 


m'z' 


-  -.-J----      (230) 
1  -j-  m        ^ 


The  equations  (15)i  are  now  easily  transformed  into  the  followiiig: — 
g.^^(L^^,.P(2^-.„)(^-i..)  (231) 

+  ^H2/o  +  'H'2/;)(i,-^). 

which  completely  determine  the  motion  of  the  comet  about  the  com- 
mon centre  of  gravity  of  tiie  sun  and  planet.  The  second  niendjers 
express  the  forces  which  disturb  the  pure  elliptic  motion ;  and  it  is 
evident,  by  an  inspection  of  the  terms,  that  when  the  comet  is  remote 
from  both  the  planet  and   the  sun  these  Corces  become  extremely 


538 


THEORETICAL   ASTRONOMY. 


>i 


snialJ.  If,  tlicrefore,  we  compute  the  porturbsitions  of  the  motiou 
relative  to  tlie  sun  as  far  as  to  the  point  at  which  the  second  nienihers 
of  (231)  have  not  any  appreciable  influence  on  the  results,  it  ^\■ill 
suffice  simply  to  convert  the  elements  which  refer  to  the  centre  of 
the  sun  into  those  relative  to  the  common  centre  of  gravity  of  t!io 
sun  c  nd  disturbing  planet,  and  then  to  regard  the  motion  as  undis- 
turbrd  until  the  comet  again  approaches  so  near  that  the  direct  per- 
turb tions  must  be  considered,  at  which  point  the  motion  will  again 
be  referred  to  the  centre  of  the  sun. 

203.  The  reduction  of  the  elements  from  the  centre  of  gravity  of 
the  sun  to  the  common  centre  of  gravity  of  tlie  sun  and  the  disturb- 
ing planet,  may  be  easily  effected  by  ineans  of  the  variations  of  the 
rectiingular  co-ordinates  and  of  the  corres])onding  velocities.  To 
derive  the  co-ordinates  of  the  comet  referred  to  the  centre  of  gravity 
of  the  sun  and  planet,  it  is  oidy  necessary  to  add  to  the  heliocentric 
co-ordinates  the  co-ordinates  of  the  sun  referred  to  this  origin,  so 
that,  according  to  (230),  we  shall  liave 


m' 


and,  also. 


dx 


oy=z  — 


in' 


1  -f  iii 


:v2/. 


m 


d^ 

1  +  m'  '  dt ' 

dz 


d~ 


dij 

'lit 


Sz 


m' 


.  «~  m'         dz' 

''dt~'~  1  +  m'  '  W 


1-f-m' 


tz',     (232) 


VI 


dj/_ 

1  -f  m'  '   dt  ' 


(233) 


If,  therefore,  from  the  elements  of  the  orbit  of  the  disturbing  j^lnnot 
we  compute  the  auxiliary  constants  for  the  adopted  fundamental 
plane  by  means  of  the  equations  (94)i  or  (99),,  and  also  V  and  U' 
from 


--=-- —  (e  sm  u>'  -f  sm  u)  =  V  sm  U  , 

Vp' 

kVl  +  m'  .  ,  ,  »^      T7/        TT' 

■ -,-=  -    (e  cos  u»  -f-  cos  «  )  =  V  cos  U  , 

Vp' 

the  equations  (100),  and  (49),  in  connection  with  (232)  and  (233), 
give 

m' 


Sx  = 


1  +  m' 


r  r'  sin  a'  sin  (A'  -f  «')> 


(234) 


PERTURBATIONS  OF  COMETS. 


539 


TO' 


dy  =  —     Jfl-,-  /  sin  h'  sin  {B'  +  ;/'), 


52=-~ 


— T  r'  sin  c'sin  (  C"  +  v!) : 

1  -f  TO  ^  I         / ' 


(it 


m 


,    ,--~Fsina'cosM'4-C7'), 
1 4-  m  ^  ^ 


(234) 


m 


1  +  «i' 


>-  F'  sin  6'  cos  (B'  +  6^'). 


1  +  m' 


rV  sine' cos  {C  -{-U'), 


dx     ^  (7;/         T    ^  dz 
o~Tz'  and  r)  ,. 

(W  dt 


to  the  cor- 


If  we  add  the  values  of  dx,  dy,  dz,  3  ,  ■ 

responding  co-ordinates  and  velocities  of  the  comet  in  reference  to 
the  centre  of  gravity  of  the  sun,  the  results  will  give  the  co-ordinates 
and  velocities  of  the  comet  in  reference  to  the  common  centre  of 
gravity  of  the  sun  and  disturbing  planet,  and  from  these  the  new 
elements  of  the  orbit  may  be  determined  as  explained  in  Art,  168. 

The  time  at  which  the  elements  of  the  orbit  of  the  comet  may  be 
referred  to  the  common  centre  of  gravity  of  the  sun  and  planet,  can 
be  readily  estimated  in  the  actual  application  of  the  formuhe,  by 
moans  of  the  magnitude  of  the  disturbing  force.  In  the  case  of  Mer- 
cury as  the  disturbing  planet,  this  transformation  may  generally  be 
effected  when  the  radius-vector  of  the  comet  has  attained  the  value 
1.5,  and  in  the  case  of  Venus  when  it  has  the  value  2.5.  It  should 
be  remarked,  however,  that  the  distance  here  assigned  may  be  in- 
creased or  diminished  by  the  relative  position  of  the  bodies  in  their 
orbits.  The  motion  relative  to  the  common  centre  of  gravity  of 
the  sun  and  planet — disregarding  the  perturbations  produced  by  the 
other  planets,  which  should  be  considered  separately — may  then  be  re- 
garded as  undisturbed  until  the  comet  has  again  arrived  at  the  point 
at  which  the  motion  must  be  referred  to  the  centre  of  the  sun,  and  at 
which  the  perturbations  of  this  motion  by  the  planet  under  consider- 
ation must  be  determined.  The  reduction  to  the  centre  of  the  sun 
will  be  effected  by  means  of  the  values  obtained  from  (234),  when  the 
second  member  of  each  of  these  equations  is  taken  with  a  contrary 


sign. 


204.  In  the  cases  in  which  the  motion  of  the  comet  will  be  referred 
to  the  common  centre  of  gravity  of  the  sun  and  disturbing  })lanet, 
the  resulting  variations  of  the  co-ordinates  and  velocities  ivill  be  so 
small  that  their  squares  and  products  may  be  neglected,  and,  there- 


540 


THEORETICAL   ASTRONOMY. 


fore,  instead  of  using  the  complete  formula)  in  finding  the  new  ele- 
ments, it  will  suffice  to  employ  diflerential  formula;.  The  formulie 
(100),  give 


dx 

'di 

dy 

dt 

dz 

W 


■  sm  a  sm  {A  -f-  u)  -%  -  -f-  r  sm  a  cos  {A  +  «)  -,t> 

:  sin  b  sin  (B  -\-  ii)  -j-  -{-  r  sin  b  cos  (B  -\-  u)  -j-, 

dv  dv 

sin  c  sin  (  (7  +  ■**)  -TT  +  »*  sin  e  cos  (  C  +  «)  -t,- 


(235) 


If  we  multiply  the  first  of  these  equations  by  Sx,  the  second  by  dy, 

dx 
and  the  third  by  dz;  then  multiply  the  first  by  ^  --->  the  second  by 


Q'V  dz 

^-jf'  and  the  third  by  ^-^>  and  put 


dt 


(236) 


we 


P=  sin  a  sin  (^4  +  u)  Sx  -f-  sin  b  sin  (B  -f  «)  dy 

-\-  sin  c  sin  (  (7  -f  u)  Sz, 
Q  =  sin  a  cos  (^  +  «)  ^x  -f-  sin  6  cos  (B  +  «)  ''^Z 

+  sin  c  cos  (  C  + 'iO  ^2 ; 

dv  dv 

P'  =  sin  a  sin  (u4  +  -«)  3~-\-  smb  sin  CJB  +  m)  d-^ 

dz 
-f-  sin  c  sin  (  C  -|-  «)  o   ,^-, 

dx  dv 

Q'  =  sin  a  cos  (^  +  «)  ^-jj  +  sin  6  cos  {B  +  «)  <?-— 

-f  sin  c  cos  (  C  +  ■'0  ^  —,j-' 
shall  have,  observing  that  -jr-  =  —7=-  e  sin  v  and  that  -,-  =  '  ,/-. 


dx  .     ,    dy  .     ,    dz  .  h        .       „  ,    A;i/m  „ 


dx   ^  dx    ,    fZw  .  dy    ,    dz  .  dz         k        .       „    ,   A-K?)  /)' 
dt      dt        dt      dt        dt     dt       i/p  r 


mi) 


From  the  equations 


dr 
It 


dx 
dt 


r—^x~-\-y-^  +  z-rr, 


dy_ 
dt 


dz 
It 


'^   ~  dt'  "^  dt'  "^  dt»' 


he  new  ele- 
he  forraulte 


'v 


It 


(235) 


Mnd  by  8y, 
!  second  by 


(236) 


iy 

dt 

dz 

It' 

dy_ 

dt 

■h 

■it' 


mi) 


IQ'. 


PERTURBATIONS  OF  COMETS. 


541 


we  get 


dt 


(it 


which  by  means  of  (237)  become 
d 


Vp 


kVp 


F<JF=-~;-esin^P'+i^^O' 


(238) 


V  p 


<2' 


From  the  equation 


we  get 


k^p  ^-  FV 


IF 


2pkdk  +  k'l^p  =  2r'  F<5  F  +  2  FVoV  —  2  '-^  <J  /  -'^'"  1 

dt     \  dt  )' 

Substituting  the  values  given  by  (238),  observing  also  thatP=.5. 
tins  becomes  ' 

5:  +  iP  ._  V'-  p      re'  sin^  v  _       e  sin  v 


k     '    2p~'k'p 
cad,  since 


P' 


P 


'^+S7«'^ 


k^ 


we  obtain 


F'  =  -  (1  4-  2e  cos  V  +  e'), 


by  means  of  which  the  variation  of  |/-  may  be  found. 
Tlie  equation 

a         r 

gives 


(239) 


a 


=  _  5^_        Tr,5p._^2 


(--'If-' 


from  which  we  derive 


fJ-  = 


P  — 


2e 


^v/ 


sm  V 


~P'~ 


P 


21/ 


P 


Q!  +  2 


[l-D 


sk 


(240) 


642 


THEORETICAL  ASTRONOMY. 


from  which  the  new  value  of  the  serai-transverse  axis  a  may  be 
found.     To  find  d/j.  we  have 


1  dk 

8n  =  lim8--\-  ii-j--, 


(241) 


or 


'S/m  p      S/Mcsmv  j^,       S/mVp 


ky/ 


P 


rk 


g+{^-2)4.  im 


Next,  to  find  3e,  we  have,  from p  =  a{\  —  <?), 


2e    a 


<Je  =  -^5--— ^-<5(y-), 


ae 


(243) 


or 


^e  =  ^—  P  +  -—  <2  H Y^'  P  +  ~-f-  (cos  V  +  cos  E)  q 


2p  cos  E     5k 


The  equation  (12)3  gives 


>"'       ^          »■'  sin  v    .„   ,  .  . 

5v :; ; —  (2  +  e  cos  v)  6e, 


a'  cos  <p  a'  COS"  <p 

and  from  -  =  l-\-e  cos  v  we  get     • 


(244) 


(245) 


-  cosv   -     ,        p 

e  sin  V  re  sm  v 


dr 


re  sm 


(246) 


Substituting  this  value  of  8v  in  (245),  and  reducing,  we  find 

,,,  I  cot d)   ,   tan  (j)  \   .       „  ,     cos?;    _    ,       1     ,        ,  ,  -  ,  „. 

oM=z  —  \ -\ ^  Ism  vP-\ Q-\ ==  (pcot<Jcos«  — 2i'cos(i)  P' 

\r^af  ^  atsmip^^  f-i/p  ^^ 

1         (p  +  r)smv^,    ./cctri    ,    tan<j\„     .       6k  (^'^'^^ 
-=  .  -^-  — ; <2   + 1  —  -  H 1 2r  sin  w  -7-, 

from  which  to  derive  the  variation  of  the  mean  anomaly. 

205.  Let  us  now  denote  by  ic",  y",  z"  the  heliocentric  co-ordinates 
of  the  comet  referred  to  a  system  in  whicli  the  plane  of  the  orbit  is 
the  fundamental  plane,  and  in  which  the  positive  axis  of  x  is  directed 
to  the  ascending  node  on  the  ecliptic.  Let  us  also  denote  by  x\  //',  2 
the  co-ordinates  referred  to  a  system  in  which  the  plane  of  the  ecliptic 
is  the  plane  of  xy,  and  in  which  the  positive  axis  of  x  is  directed  to 
the  vernal  equinox.     Then  we  shall  have 


PERTURBATIONS  OF  COMETS. 


543 


a/'  =  «'  cos  SI  +  y'  sin  SI , 

y"  =  —  x'  sin  J^  cos  i  +  y'  cos  J^  cos  i  +  s'  sin  i, 

a"  =  a/  sin  ft  sin  i  —  y'  cos  ft  sin  i  +  z'  cos  i, 

If  we  transform  the  co-ordinates  still  further,  and  denote  by  .r,  y,  z 
the  co-ordinates  referred  to  the  equator  or  to  any  other  plane  nuiking 
the  angle  e  with  the  ecliptic,  the  positive  axis  of  x  being  directed  to 
the  point  from  which  longitudes  arc  measured  in  this  plane;  and  if 
we  introduce  also  the  auxiliary  constants  a.  A,  h,  B,  &c.,  we  shall 
have 

dx"  =  sin  a  sin  A  ox  -f  sin  h  sin  B  Hy  -\-  sin  c  sin  C  8z, 

8y"  =  sin  a  cos  A  <^x  +  sin  b  cos  B  5y  -\-  sin  c  cos  C  Sz,        (248) 

S^'  =  cos  aSx-\-  cos  b  dy  -\-  cos  c  5s. 

Multiplying  the  first  of  these  by  —  sin  u,  and  the  second  by  cos  u, 
adding  the  results,  and  introducing  Q  as  given  by  the  second  of 
equations  (236),  we  get 

cos  u  dy"  —  sin  u  Sx"  =  Q. 

Substituting  for  dx"  and  8y"  the  values  given  by  the  equations  (73)o, 

the  result  is 

r  (dv  +  Sx)  ^  Q, 

and,  introducing  the  value  of  dv  given  by  (246),  we  obtain 


r       e  sm  V  r'e  sin  v 


^r  +  -:^^HVp)- 


re  sm  i 


Substituting  further  for  de,  dr,  and  d{i/p)  the  values  already  ob- 
tained, and  reducing,  we  find 


sin  v  cos  E  cos  vVp  „,   ,   (|)  +  r)  sin  v  ^ 

2  sin  V     ^k 


(249) 


by  means  of  which  b^  may  be  found. 
If  we  put 

cos  a  5*  +  cos  b^y  •\-  cos  c  5z  =  22, 

, ttr    ,         1  ^dy    .  .dz         . 

cos  a  5-rr  +  cos  b  S-jt-  -{■  cos  c  o—-  -.^K . 
dt  dt  ct 

the  last  of  the  equations  (248)  gives 


(250) 


/ 


544  THEORETICAL  ASTRONOMY. 

Sz"  =  R;  ■  (251) 

and  if  wc  differentiate  the  equation 

dz    ,         ,  dy    ,  rfz       ^ 

cos  a-TT  -f  cos  0 -jf  +  cos  c-,-  =  0, 
dt  dt  dt 

which  exists  in  the  case  of  tlie  unchanged  elements,  we  shall  have 

A  'sdx    .         I  ^dy    ,  ^dz 

0  =  cos  a  o-r-  -f  cos  0  d—f-  +  cos  c  8—r- 

dt  dt  dt 

dx    .        ,         dy    .    ,  ^,        dz    . 

j7-  sin  aoa -■  sin  bSb rr  sin  c  3c. 

dt  dt  dt 

Substituting  for  na,  oh,  and  dc  the  values  given  in  Art.  60,  observing 
that  dz  =  0,  we  have 


0=iJ'  +  l    .rsinasin^  -{■-J-smhsmB -{•  -TrsincsinClsini  ^J^ 


-I  -jp  sin  a  cos ^  +  -Jr  sin  6  cos  -B  +  --^j  sin  c  cos  C  J  3i. 


(252) 


From  the  equations  (100)i,  observing  that  the  relations  between  the 
auxiliary  constants  are  not  changed  when  the  variable  u  is  put  equal 
to  zero,  or  equal  to  90°,  we  get 


sin*  a  sin'  A  +  sin'  b  sin'  B  +  sin'  c  sin'  C^^l, 
sin'  a  cos'  A  +  sin'  b  cos'  B  +  sin'  c  cos'  C      1, 


(253) 


and  from  (235)  wc  find 

sin'  a  sin  J.  cos  -4  -f  sin'  b  sin  B  cos  B  +  sin'  c  sin  Ccos  C=  0.     (254) 

Substituting  in  (252)  for     ,. »    /'  and   ^   the  values  given  by  the 
equations  (49),  and  reducing  by  means  of  (253)  and  (254),  we  get 

0  =  i?'  —  FsinO'sin  i  5£l—VcosU8i.  (255) 

Substituting  further  for  dz"  in  (251)  the  value  given  by  the  last  of 
the  equations  (73)2,  there  results 


0  =  R  -{•  r  cos  %!.  sin  i  8Q,  —  r  sin  -u  Si 
From  these  equations  we  derive,  by  elimination, 


(256) 


PEUTURUATK^XS   OF   COMETS. 

rsin  (( 


am  I 


,  _  e  009  lu  -j-  cos  II  n   ,      1 

ssi=  —  — — A-. E  H — ;- 

P  «>"  «  kV  p 

,.              e  sin  ut  -f-  sin  «  _,   ,  7'  cos  u  ,., 
01  =  li  -+-  —    ---  ii , 


/?', 


645 


(257) 


by  means  of  which  dQ  and  oi  may  be  found.     To  find  o(o  and  or  we 

have 

du}  =  dx  —  cos  i«5 ft ,  o;r  =  J;/  +  2  sin»  \  io  ft ,  (258) 

(ly  being  found  from  equation  (249). 

Neglecting  the  mass  of  the  comet  as  inappreciable  in  coini)arison 
with  that  of  the  sun,  the  attractive  force  which  acts  upon  the  comet 
in  the  case  of  the  undisturbed  motion  relative  to  the  sun  is  k^,  but  in 
the  case  of  the  motion  relative  to  the  common  centre  of  gravity  of 
the  sun  and  jdanet  th's  force  is  Ir  {1 -\- m').  Hence  it  follows  that 
the  increment  of  this  force  will  be  la'k^,  and  we  shall  have 


Sk 


=  Im', 


(259) 


by  means  of  which  the  value  of  this  factor,  which  is  required  in  the 
formula}  for  3{y^p),  5  ->  tfcc,  may  be  found. 

206.  The  formula)  thus  derived  enable  us  to  effect  the  required 
transformation  of  the  elements.     In  the  first  place,  we  compute  the 

dx     ,  dy         ,    ^  dz 


values  of  dx,  oij,  8z,  d 


dt'  'P  ""^  'm 


by  means  of  the  formuhe 


(234) ;  then,  by  means  of  (236)  and  (250),  we  compute  P,  Q,  It,  P', 
Q',  and  W ,  the  auxiliary  constants  a,  A,  &q.  being  determined  in 
reference  to  the  fundamental  ])lane  to  which  the  co-ordinates  are  re- 
ferred. AVlien  the  fundamental  ]>lane  is  the  plane  of  the  ecliptic,  or 
that  to  which  ft  and  i  arc  referred,  we  have 


sm  c  =  sm  ?, 


C=0. 


The  algebraic  signs  of  cos  a,  cos  6,  and  cose,  as  indicated  by  the  equa- 
tions (101),,  must  be  carefully  attended  to.  The  formuhe  for  the 
variations  of  the  elements  will  then  give  the  corrections  to  be  ap})lied 
to  the  elements  of  the  orbit  relative  to  the  sun  in  order  to  obtain 
those  of  the  orbit  relative  to  the  common  centre  of  gravity  of  the 
sun  and  planet.  Whenever  the  elements  of  the  orbit  about  the  sun 
are  again  required,  the  corrections  will  be  determined  in  the  same 
manner,  but  will  be  applied  each  with  a  contrary  sign. 

35 


546 


TIIKOUKTICAL   AST1{(»\<)MY. 


Sinco  the  equations  liuve  l)een  derived  for  the  viiriations  of  more 
than  the  six  eU'inents  usually  eniph)yed,  the  acMitionul  fonuuhe,  as 
well  as  those  which  j^ive  ditlereiit  relations  between  the  elements  em- 
ployed, may  be  used  to  eheek  the  numerietil  ealeulation;  and  this 
proof  should  not  be  omitted.  It  is  obvious,  also,  that  these  differen- 
tial formulic  will  serve  to  convert  the  perturbations  of  the  rectaufrular 
eo-ordinates  into  jx'rturbations  of  the  elcnnents,  whenever  the  terms 
of  the  second  order  may  be  neglected,  observing  that  in  this  case 
t?/;  =  0.  If  some  of  the  elements  considered  are  expressed  in  angular 
measure,  and  some  in  parts  of  other  units,  the  quantity  8=^  200204". 8 
should  be  introduced,  in  the  numerical  application,  so  as  to  preserve 
the  homogeneity  of  the  formulse. 

AVhen  the  motion  of  the  comet  is  regarded  as  undisturbed  about 
the  centre  of  gravity  of  the  system,  the  variations  of  the  elements  for 
the  instant  t  in  order  to  reduce  them  to  the  centre  of  gravity  of  the 
system,  added  algebraically  to  those  for  the  instant  i'  in  order  to 
reduce  them  again  to  the  centre  of  the  sun,  will  give  the  total  ]u  rtur- 
bations  of  the  elements  of  the  orbit  relative  to  the  sun  duri  the 
interval  t'  —  t.  It  should  be  ob.served,  however,  that  the  value  of 
oJ/  for  the  instant  t  should  be  rechiced  to  that  for  the  instant  t',  so 
that  the  total  variation  of  il/ during  the  interval  t'  —  t  will  be 

In  this  manner,  by  considering  the  action  of  the  several  disturbing 
bodies  separately,  referring  the  motion  of  the  comet  to  the  common 
centre  of  gravity  of  the  sun  and  })lauet  whenever  it  may  subsequently 
be  regarded  as  undisturbed  about  this  point,  and  again  referring  it  to 
the  centre  of  the  sun  when  such  an  assumjition  is  no  longer  admissi- 
ble, the  determination  of  the  perturbations  during  an  entire  revolu- 
tion of  the  comet  is  very  greatly  facilitated. 


207.  If  we  consider  the  position  and  dimensions  of  the  orbits  of 
the  comets,  i^  Mill  at  once  appear  that  a  very  near  approach  of  some 
of  these  bodies  to  a  planet  may  often  happen,  and  that  when  they 
ajjproach  very  near  some  of  the  large  planets  their  orbits  may  be 
entirely  changed.  It  is,  indeed,  certainly  known  that  the  orbits  of 
comets  liavc  been  thus  modified  by  a  near  approach  to  Jupiter,  and 
there  are  periodic  comets  now  known  which  will  be  eventually  thus 
acted  upon.  It  becomes  an  interesting  problem,  therefore,  to  con- 
sider the  fonnulaj  applicable  to  this  special  case  in  which  the  ordinary 
methods  of  calculating  perturbations  cannot  be  applied. 


rERTITllBATlON.S   OF   COMKT8. 


647 


s  of  more 
nniilii',  as 
lU'iil.s  cin- 

and  this 
)  (liffm'on- 
ictiuigular 
tlie  tonus 

this  case 
n  anguhir 
:062G4".8 
)  prt'sorvo 

)cd  about 
imcnts  for 
ity  of  the 

order  to 
al  ]»'  rtnr- 
iiri       tlie 

value  of 
taut  t',  so 
be 


listurbiiig 
)  common 
•sequontly 
•ring  it  to 
•  admissi- 
•e  revohi- 


orbits  of 
li  of  some 
,'hen  tliey 
s  may  be 

orbits  of 
pitcr,  and 
lally  thus 
3,  to  eon- 
2  ordinary 


If  we  denote  by  .r',  //',  s',  v',  the  co-ordinates  and  radius-vector  of 
the  ])huiet  referred  to  the  centre  of  the  sun,  and  reiiard  its  motion 
rehitive  to  the  sun  as  disturbed  by  the  comet,  we  shall  have 


(If 

+ 

(I'll' 
d'l' 

+ 

dh' 
df 

+ 

P(l  -^  m')z' 

«.fL;?-,^),  (200) 


m 


mk 


Let  us  now  denote  by  ^,  r^,  ^  th(!  co-ordinates  of  the  comet  referred 
to  the  centre  of  gravity  of  the  planet;  then  will 


y  —  y> 


:  =r-  3  —  z'. 


Substituting  the  resulting  values  of  x' ,  y' ,  z'  in  tlio  preceding  equa- 
tions, and  subtracting  these  from  the  corresponding  equations  (1)  for 
the  disturbed  motion  of  the  comet,  we  derive 


7? 


df  "^     '  f>'      ~^  \  r'-'      ?  r 

A       k'  (m  +  m'  )r,__,J][^      y'±-'i\ 
"df  +  ?'  ^ "  \  >"  r''     /' 


(2G1) 


d'" 
dt 


1  + 


k'  (m  4-  m")  % 


^'\?^  ,,3     )• 


These  ecpiations  express  the  motion  of  the  comet  relative  to  the  centre 
of  gravity  of  the  disturl)ing  planet;  and  when  the  comet  approacb.es 
very  near  to  the  planet,  so  that  the  second  mend)er  of  each  of  these 
equations  becomes  very  small  in  comparison  with  the  second  term 
of  the  first  member,  we  may  take,  ibr  a  first  approximation, 


df'^  (>'  --^' 

iVt)       li-  (m  +  '»;.')  t;  __ 
rf?  "^  f>'  ~ "' 

dK      />■'  (m  +  m')  :  _ 


(262) 


and,  since  ~ ; is  the  sum  of  the  attractive  force  of  the  planet 

on  the  comet  and  of  the  reciprocal  action  of  the  comet  on  the  planet, 


548 


TIIEORETirAL    ASTIIOXOMY. 


( 


tlioMc  c(|iirttions,  bcinjx  of  the?  saiiic  form  as  those  for  tlio  undisturlxd 
iimtioii  of  lilt!  comet   relative  to  tlu!  sun,  show  that  when  tlie  aetimi 
of  the  (listiu'hini^  planet  on  tlio  comet  exceeds  that  of  tlie  sun,  the 
result  of  the  first  approxinmtion  to  the  motion  of  the  comet  is  that 
it  (IcscribcH  a  conic  section  around  the  centre  of  f:;ravity  of  the  phmet. 
Further,  since  — x',  — y',  — z'  are  the  co-ordinates  of  the  sun  re- 
ferred  to  the  centre  of  jijravity  of  the  phmet,  it  apj)ears  that  tlic 
second  memhers  of  (201)  express  the  disturbing  force  of  the  sun  (ui 
the  comet   resolved   in  dire<!tions    parallel    to  the  co-ordinate  axes 
respeetiv(!ly.     Hcim;  »I'.en  a  comet  approaches  so  near  a  planet  that 
tlie  action  of  the  latter  upon  it  exceeds  that  of  the  sun,  its  motion 
■will  be  in  a  conic  section  relatively  to  the  planet,  and  will  be  dis- 
turbed by  the  action  of  the  sun.     But  the  disturbing  action  of  the 
sun  is  the  difference  between  its  action  on  the  comet  and  on  the 
planet,  and  the  masses  of  the  larger  bodies  of  the  solar  system  are 
such  that  when  the  comet  is  equally  attracted  by  the  sun  and  by  the 
l)lanet,  the  distances  of  the  comet  and  planet  from  the  sun  ditler  so 
litthi  that  tlie  distur])ing  force  of  the  sun  on  the  comet,  regarded  as 
describing  a  conic  section  about  the  planet,  will  be  extrenudy  small. 
Thus,  in  a  direction  parallel  to  the  co-ordinate  f  the  disturbing  force 
exercised  by  the  sun  is 


^\r''  --,^-')-^"(>— ^^  )' 


and  when  the  comet  api)roaclies  very  near  the  planet  this  force  will 
he  extremely  small.  It  is  evident,  further,  that  the  action  of  the 
sun  regarded  as  the  disturbing  body  will  be  very  small  even  when 
its  direct  action  u])on  the  comet  considerably  exceeds  that  of  the 
planet,  and,  therefore,  that  we  may  consider  the  orbit  of  tlie  comet  to 
be  a  conic  section  about  the  planet  and  disturbed  by  the  sun,  when  it 
is  actually  attracted  more  by  the  sun  than  by  the  planet. 

208.  In  order  to  show  more  clearly  that  the  disturbing  force  of  the 
sun  is  very  small  even  when  its  direct  action  on  the  comet  exceeds 
that  of  the  })lanet,  let  us  suppose  the  sun,  planet,  and  comet  to  be 
situated  on  the  same  straight  line,  in  which  case  the  disturbing  force 
of  the  sun  will  be  a  maximum  for  a  given  distance  of  the  comet  from 

the  planet.     Then  will  the  direct  action  of  the  sun  be  -^,  and  that 

of  the  planet  —  j-*      The  disturbing  action  of  the  sun  will  be 


PERTrKHATIOXS   OF  COMKTS. 


k'n      2r  -h  p 


548 


{r-\-f>)'        r'      (/•  +  /'/ 


wlik'li,  since  f>  is  suppu.scd  to  bo  small  in  loinpHrisDu  with  v,  may  ba 
put  I'quiil  to 

2k'p 


and  Iicnt'c  the  ratio  of  the  di.stnrbin};  action  of  the  sun  to  the  direct 
action  of  ti»e  planet  on  the  ('()mct  cannot  exceed 


/?  = 


in  f 


If  the  comet  is  at  a  distance,  such  that  the  direct  action  of  the  sun  is 
equal  to  the  direct  action  of  the  planet,  we  have 

p^  =  «i'»*', 

and  the  ratio  of  the  direct  action  of  the  siui  to  its  disturbing  action 

cannot  in  this  case  exceed  2]^vi'.     In  the  ease  of  Jupiter  this  amounts 

to  only  0.06. 

So  long  as  ft  is  small,  the  disturbing  action  of  the  planet  is  very 

m'k^ 
nearly  -  ,^  -  in  all  positions  of  the  comet  relative  to  tlie  planet,  and 

hence  the  ratio  of  the  disturbing  action  of  the  planet  to  the  direct 
action  of  the  sun  cannot  exceed 


B' 


m  r 


At  the  point  for  which  the  value  of  p  corresponds  to  RR',  the 
coinct,  sun,  and  planet  being  supposed  to  be  situated  in  the  same 
straight  line,  it  will  be  immaterial  whether  we  consider  the  sun  or 
the  planet  as  the  disturbing  body;  but  for  values  of  ()  less  than  this 
R  will  be  less  than  R',  and  the  i)lanct  must  be  regarded  as  the  con- 
trolling and  the  sun  as  the  disturbing  body.  The  sui)positi()n  that 
R  is  equal  to  R'  gives 

2,o« 


mV 


?»'?•' 


and  therefore 


5  /    1  In 

p  =  rv  \m  , 


(203) 


Hence  we  may  compute  the  perturbations  of  the  comet,  regarding 
the  planet  as  the  disturbing  body,  until  it  aj)proaches  so  near  the 


ooO 


TII  i:<ll{ ETK'A  I.    ASTHOX(  ).M  Y. 


r ; 


:r^ 


))l:uict  tliiit  //  lia.-;  tin-  value  givi'U  by  this  ('(jitation.  after  wlueli,  sn 
loiii:'  as  (>  <l()e<  iidt  exeeed  the  vahie  here  assigned,  the  tiun  must  he 
reganh'd  as  the  (lisii.rhlug  hmly, 

if  (.'  rejtresonts  (he  angle  at  tlio  ])hinet  between  the  sun  and  eonut, 
the  disturbing  Ibrec;  oi"  the  sun,  for  any  position  ol"  the  eoinct  near 
the  ])hinet,  Avill  l)e  very  nearly 

and  when  tliis  angle  is  considerable,  the  disturbing  aetion  oi'  the  sini 
\\ili  l>e  small  even  when//  is  greater  than  ri  Ini'-,  Henee  we  niav 
eommenee  to  consider  the  sun  as  tlui  disturbing  body  even  before  the 
comet  reaches  the  point  ibr  which 


and,  sine(>  th(>  ratio  of  the  distm'bing  a<'tion  of  the  planet  to  the 
direct  action  of  the  sun  remains  nearly  the  same  tor  all  values  ot'  t's 
wiien  !>  is  within  the  limits  liero  assigned  tli(  sun  must  in  all  e;i-ts 
be  so  consider(>(l.  (Vorresponding  to  the  value  of  (>  given  by  ecpiatioii 
(20']),  we  have 

and  in  the  case  of  a  near  a])proach  to  Ju{)iter  th(!  results  are 

,>=--^)M\r,  i?':^.  0.:i3. 

209.  In  tlie  actual  calculation  of  the  perturbations  of  any  particu- 
lar comet  when  very  near  a  large  ]>lanet,  it  will  be  easy  to  determine 
the  point  at  wliicli  it  will  lie  advantageous  (o  commence  to  regard  the 
sun  as  the  disturbing  body;  and,  having  i'oun.l  the  elementt' of  the 
orbit  of  the  cotnet  relative  to  tlie  planet,  tin-  perturitations  of  these 
elements  or  of  the  co-ordinates  will  b"  obtaii  ed  by  means  (  I'  the 
Ibrmulic  already  derived,  the  aoeessary  ilistinctions  being  made  in  tlu' 
nottition.  When  the  ])lanet  agi.in  liecotnes  the  disturbing  body,  the 
elements  will  be  found  in  reference  to  the  sun;  and  thus  we  an 
enal.>led  to  t>'aee  the  motion  of  the  comet  before  and  subscMjuei.'  to  it.- 
being  considered  as  subject  principally  to  the  ))lanet.  In  the  case  of 
the  first  trai^sformation,  the  co-ordinates  and  \eloeities  of  the  comet 
and  plaiK't  in  reference  to  tlu>  sun  being  determined  for' the  ir  taut  at 
uhieh  the  sun  is  regarded  as  ceasing  to  be  the  controlling  body,  we 
shall  have 


i>Er;TT:ni'..\T[ONs  of  comets. 


551 


wlncli,  so 
1  mu>t   lie 

11(1  COllU't, 

;iuct  near 


r_r-  X  ■ 


Z=^z- 


df 


dx 

dt 


dx' 

dt' 


dr, 

'/,'/ 

<l!l' 

d: 

dz        dz' 

dl  ~'^ 

dt 

-  dt' 

dt 

~  dt        dt 

and  from  ?,  Tj 


re 


d:;      dr;  i    '^'■»      i         i  ,.    i  i  •       p    i 

,  -i  'Tr>     i/'  'inu     ,,  '  the  cIcMncnts  oi  tlie  orl)it  of  tlio 
'   • '   dt      dt  (It 

coinct  about  the  j)Iiinot  are  to  he  (leterniined  precisely  a.s  the  elements 

ill  reference  to  the  sun  are  ibuiul   from  ,r,  (/,  ,:,     "   »     '  .  and     ," .  and 

■  (It      (It  dt 

as  explained  in  ^Vrt.  108.  Jlasing  eomjinted  the  perturbations  of 
the  motion  I'elative  to  the  planet  to  the  [loint  at  wlii'  h  the  planet  is 
UL!,ain  considered  as  the  disturbing  l)ody,  it  only  remains  to  llnd,  titr 
the  eorrespoudint!;  time,  the  co-ordinates  and  velocities  of  the  comet 
in  reterence  to  the  centre  of  <i-ravity  of  the  {)!anet,  and  I'rom  these  the 
n)-ordi nates  anil  velocities  relative  to  the  centre  of  the  sun,  and  the 
elements  of  the  orbit  about  the  sun  may  be  determined.  As  the  in- 
terval of  time  duriiiu-  which  the  sun  will  be  regarded  as  the  disturb- 
ing body  will  always  be  small,  it  will  be  most  convenient  to  compute 
the  perturbations  of  the  rectangular  co-ordinates,  in  which  case  the 

\'d;;es  of  ?,  -j,  ^,     ."  <  ■,'>  and      '  will  be  obtained  directly,  and  then, 

having  found  the  corresponding  co-ordinates  x',  y',  z'  and  velocities 

dx'    dy'    dz'     „    ^        ,  .        ,,  ,  , 

,  >  -.; )  -.7-  01  the  planet  ni  reterence  to  tlie  sun,  we  have 


ly  particu- 
d(  teniiiiie 
r(\ii'ar(l  the 

•Ml.--  Ill"   till' 

|l-.  (it  llioe 
liiis  (  I'  the 
liade  ill  thi' 
body,  the 
Ins  we  ai' 
iuei:'  til  it- 
he  ease  ut 
|*tlie  comet 
ii'  taut  ai 
|;  body,  \\r 


x  —  x  +  i;, 

dx rfj;'    ._  d? 

lit  "'  (It  "''  "(It' 


dti 

dt 


dl,    dri 

dt  ~^'  dt ' 


3  =-  2'  +  ?, 

dz  _  dz'      d: 
(It^  dt'^df 


l)v  means  of  which  the  elements  of  the  orliit  relative  to  the  sun  will 
he  found.     If  it  is  not  considered  necessary  to  compute  rigorously 


tli^'  path  of  the  comet  liefore  and  after  it  is  subject  principally  (o  tin 
action  of  the  {)laiiet,  but  simply  to  llnd  the  ])rincipal  effect  of  tin 


I  artioii  of  the  planet  in  changing  its  eh'inents,  it  will  be  suilicicnt, 
iluriiig  the  time  in  which  the  sun  is  regarded  as  the  disturbing  body, 
to  ^u])pose  the  comet  to  move  in  an  undisturl)ed  orbit  abinit  the 
planet.  For  the  point  at  which  we  cease  to  regard  the  sun  a^  the 
ilisturbinsr  body,  the  co-ordinates  and  velocities  of  the  comet  relative 
to  the  centre  of  gravity  of  the  planet  will  be  determined  from  the 
I'leaients  of  the  orbit  in  reference  to  the  ])lanet,  pri'cisely  as  the  corre- 
sponding ([(lantities  are  determined  in  the  case  of  the  motion  relative 
to  the  sun,  the  necessary  distinctions  being  made  in  tiie  notation. 


552 


THEORETICAL   ASTRONOMY. 


210.  The  results  obtained  from  the  observations  of  tlic  periodic 
eoniets  at  their  sueeessive  returns  to  the  })criholion,  render  it  probable 
that  there  exists  in  s})ace  a  resisting  medium  which  opposes  the  motion 
of  all  the  lieavcnly  bodies  in  their  orbits;  but  since  the  observations 
of  the  planets  do  not  exhibit  any  effect  of  such  a  resistance;  it  is  in- 
ferred that  the  density  of  the  ethereal  iluid  is  so  slight  that  it  can 
have  an  appreciable  etlect  only  in  the  case  of  rare  and  attennatcd 
bodies  like  the  comets.  If,  however,  we  adopt  the  hypothesis  of  a 
resisting  medium  in  space,  in  considering  the  motion  of  a  heavenly 
body  we  simply  introduce  a  new  disturbing  force  acting  in  the  direc- 
tion of  the  tangent  to  the  instantaneous  orbit,  and  in  a  sense  contrary 
to  that  of  the  motion.  The  amount  of  the  resistance  will  depend 
chietly  on  the  density  of  the  ethereal  fluid  and  on  the  velocity  of  the 
body.  In  accordance  with  what  takes  place  within  the  limits  of  our 
observation,  we  may  assume  that  the  resistance,  in  a  medium  of  con- 
stant density,  is  proportional  to  the  srpiare  of  the  velocity.  The 
density  of  the  fluid  may  be  assuin;xl  to  diminish  as  the  distance  from 
the  Sim  increases,  and  hence  it  may  be  expressed  as  a  function  oi'  the 
reciprocal  of  this  distance. 

I^ot  (Is  he  the  element  of  the  path  of  the  body,  and  r  the  I'adius- 
vector;  then  will  the  resistance  be 


T- 


Ml)%' 


(2(34) 


K  being  a  constant  quantity  depending  on  the  nature  of  the  body, 

and  ell  the  density  of  the  ethereal  fluid  at  the  distance  r.     Since 

the  force  acts  only  in  the  plane  of  the  orbit,  the  elements  which  de- 
fine fliC  position  of  this  })lane  will  not  be  changed,  and  hence  we  have 
oidv  to  determine  the  variations  of  the  elements  Jf,  c,  a,  and  y.  If 
Ave  denote  by  ^''y  the  angle  which  the  tangent  makes  with  the  prolon- 
gation of  the  radius- vector,  the  components  11  and  8  will  be  given  by 


and,  since 

Fogs  4'^  ■ 
we  have 


li  =  T  cos  (,'-0 


-  r.  sni  V, 


S: 


F=inv''o 


Tsin^'v 


kVp 

_..  — , 

r 


ds 


B-=-K,{l) 


e  sm  r 


\  p 


dt' 


*=->^(l)^'. 


rf8 
dt' 


(205) 


RESISTING  MEDIUJI   IN   SPACE.  553 

Substituting  tlicso  values  of  7^  and  S  in  the  equation  (205),  it  reduces  to 

edx  =^  —  2AV  !      I  .sin  v  ds. 


Now,  since 


we  have 


F=^4    ^  +  2e  cos  i»  +  e')~, 
Vp 


ds  =  Vdt  =:  —  (1  +  2c  COS  u  +  e'y-dv, 


and  hence 


e4  = 


P 


(i)-ni 


+  2e  cos  V  +  e^  -  sin  v  dv.         (2G6) 


If  we  supp'jr^e  the  function 


m^ 


K<p[-)r{l-\-2eco3v-\'C'f, 

the  \;>i'  V  of  which  is  always  positive,  to  be  developed  in  a  scries 
arranged  in  reference  to  the  cosines  of  v  and  of  its  multiples,  so  that 
we  have 

A>  (-)>•'(!  +  2e  cos  v  +  e'f  ^  A -{- B  cos  v -\- C  cos  2v -\-  &c.,    (267) 

in  wliich  A,  B,  Sic.  are  positive  and  functions  of  c,  the  equation  (266) 

becomes 

2 
edx  ^^ ( A  -\-  B  cos  V  +  •  •  •  .)  sin  v  dv. 


Hence,  by  integrating,  wo  derive 
2 


eS-/v=—  {A  cos  V  -f-  \  Bcos,2v  -r  .  .  .  .)> 


(268) 


from  which  it  appears  tliat  y  is  subject  only  to  periodic  perturbations 
on  account  of  tlie  resisting  nu-Alntn. 

In  it  <4milar  manner  it  may  be  nli^wn  that  the  second  term  of  the 
second  member  of  equation  (210j  profluces  only  jKirifxlic  terms  in  the 
value  (»f  fiM,  so  that  if  we  seek  only  the  s«'ular  jn-rturbations  due  to 
the  action  of  the  ethereal  fluid,  the  fifjit  and  wn.'ond  t«>rms  of  tlie 
second  m»»mber  of  (210)  will  not  r>e  considered,  and  only  the  soculiu' 
perturbatio**'  arising  from  tlw  variation  (.A  //  will  bi-  re(|uinHl. 

Let  us  litiext  consider  the  Acmuuia  a  »ad  e.     Substituting  in  the 


554 


THEORETICAL   ASTRONOMY. 


equations  (198)  and  (202)  the  values  of  i?  and  /S' given  by  (265),  and 
reducing,  we  get 


da 
de 


=  —  ^V^(  i  )  rHl  +  2e  cos  v  +  e")  ^dv, 

2        /  1  \  .' 

= K(p  I  —  I  r'  (1  +  2e  cos  u  -|-  e°)  ?  (e  +  cos  v)  dv. 


(269) 


If  we  introduce  into  these  the  series  (2G7),  and  integrate,  it  will  he 
found  that,  in  addition  to  the  periodic  terms,  the  expressions  for  oa 
and  oc  contain  each  a  term  multiplied  by  v,  and  hence  increasing  with 
the  time.  It  is  to  be  observed,  furtliei',  that  since  A  and  B  are  posi- 
tive, the  secular  variation  of  «,  and  also  that  of  c,  will  be  negative, 
and  hence  the  resisting  medium  acts  continuously  to  diminish  both 
the  mean  distance  and  the  eccentricity. 

211.  The  magnitude  of  the  disturbing  force  ai'ising  from  the  action 
of  the  resisting  medium  is  so  small  that  the  periodic  terms  have  no 
sensible  influence  on  the  place  of  the  comet  during  the  period  in 
which  it  may  be  observed;  and  hence,  since  the  effect  of  the  resist- 
ance will  be  exhibited  only  by  a  comparison  of  observations  made  at 
its  successive  returns  to  the  perihelion,  the  effect  of  the  planetary  per- 
turbations being  first  completely  eliminated,  it  is  only  necessary  to 
consider  the  secular  variations.  Further,  since  "^  is  subject  only  to 
periodic  changes  in  virtue  of  the  action  of  the  resistance,  and  since 
the  mean  longitude  is  subjected  to  a  secular  change  only  through  n, 
it  will  suffice  to  employ  the  formula;  for  d^  and  de  or  8ip.  The 
variations  of  these  elements  may  be  computed  most  conveniently  by 

mechanical  quadrature  from  given  values  of  -;  and  --,,  or  ~y  ,  al- 
though their  values  for  one  complete  revolution  of  the  comet  may  he 
determined  directly,  the  values  of  the  coefficients  A  and  B  wliich 
ap])ear  in  the  series  (267)  being  found  by  means  of  elliptic  fiuictKnis. 
The  calculation  of  the  effect  of  the  resisting  medium  will  be  n\ade  in 
connection  with  the  determination  of  the  planetary  perturlwtions.  so 
that  there  will  be  no  inconvenience  in  adding  to  the  results  ^'^e  tjc* m^ 
depending  on  this  resistance.     Since 


dji 
It 


3  m 
2  a 


da 
It' 


d(p 
W 


dc 


the  equations  (269)  give,  putting  K—  h^U, 


EESISTING   MEI)IUJ[   IN  SPACE. 


555 


'dt 
dtp 
~dX 


r  cos  <p  \  r  J 


(270) 


It  remains  now  to  make  an  assumption  in  regard  to  the  law  of  the 
density  of  the  resisting  medium.  In  the  case  of  Encke's  comet  it 
has  been  assumed  that 


il)=-^' 


and  this  hypothesis  gives  results  which  s'  fficc  to  represent  the  obser- 
vations at  its  successive  returns  to  the  perihelion.  Substituting  for  V 
its  value  in  terms  of  r  and  a,  the  equations  (270)  thus  become 


dt  r^  \  r  a  I 


d/j. 
~di 
d<p 
~di 


-^V-=:  — 2ifc'!7 


a  cos  <p  cos  E 


(l-^F' 


(271) 


by  means  of  which  on  and  d(p  may  be  found ;  and  from  any  given 
value  of  ofx  we  may  derive  the  corresponding  value  of  oa.  The 
variation  of  M,  neglecting  the  periodic  terms  arising  from  the  first 
and  second  terms  of  the  second  member  of  equation  (210),  will  be 
given  by 

which  will  be  integrated  by  mechanical  quadrature  so  as  to  include 
the  interval  of  an  entire  revolution  of  the  comet.  The  quantity  U 
has  been  determined,  by  means  of  observations  of  Encke's  comet,  to  be 


U= 


894.892 


This  value  may  be  corrected  by  introducing  a  term  in  the  equations 
of  condition  precisely  as  in  the  case  of  the  determination  of  the  cor- 
rection to  be  applied  to  the  mass  of  a  disturbing  planet.  Intro- 
ducing U  into  the  equation  (2G4),  and  adopting  the  hypothesis  tliat 

y'(      |  =  -j>  the  expression  for  the  action  of  the  ethereal  tluid  be- 


comes 


T^-lVy,, 


556 


THEORETICAL   ASTRONOMY. 


Since  tlio  constant  L^'dopencls  on  the  nature  of  the  comet,  the  value 
obtained  in  tlie  ease  of  Encke's  coniot  may  be  very  different  from 
that  in  tlie  case  of  another  comet.  Thus,  in  the  ease  of  Faye's  comet 
the  value  has  been  found  to  be 


?7  = 


1      . 

10.232' 


and  in  tlie  aj)plication  of  the  formuhc  to  the  motion  of  any  particular 
body  it  will  be  necessary  to  nudce  an  independent  determination  of 
this  constant. 

212.  The  assum])tion  that  the  density  of  the  ethereal  fluid  varies 
inversely  as  the  scpiarc  of  the  distance  from  the  sun,  is  that  which 
appears  to  be  the  most  probable,  and  the  results  obtained  in  accoi'd- 
ancc  therewith  seem  to  satisfy  the  data  furnished  by  observation.  It 
is  true,  however,  that  the  whole  subject  is  involved  in  great  uncer- 
tainty as  regards  the  nature  of  the  resisting  medium,  so  that  the 
results  obtained  by  means  of  any  assumed  law  of  density  arc  not  to 
be  regarded  as  absolutely  correct. 

From  the  formuhc  which  have  been  given,  it  appears  that,  whatever 
may  be  the  law  of  the  density  of  the  resisting  fluid,  the  mean  motion 
is  constantly  accelerated  and  the  eccentricity  diminished,  and  we  may 
determine,  by  means  of  observations  at  the  successive  appearances  of 
the  comet,  the  amount  of  these  secular  changes  independently  of  any 
assumi)tion  in  regard  to  the  density  of  the  ether.  ]^ct  x  denote  the 
variation  of  //  during  the  interval  r,  which  may  be  approximately  the 
time  of  one  revolution  of  the  comet,  and  let  y  denote  the  correspond- 
ing variation  of  (p;  then,  after  the  lapse  of  anv  interval  t  —  7^,  we 
shall  have 


.t-To 


P  =  V'o  4- 


t-i: 


'y> 


(272) 


ami,  since  the  average  variation  of  //  during  the  interval  t  — 1[,  is 


t~Z 


¥-^^. 


i»/=j»4  +  /x„(<-!r„)  + 


(t-ny 


(273j 


If  we  introduce  x  and  y  as  unknown  quantities  in  the  equations  of 
condition  for  the  correction  of  the  elements  by  means  of  the  ditll  r- 
cnoes  between  c-  niputation  and  observation,  the  secular  variations  of 
fi.  and  ^  may  be  determined  in  connection  with  the  corrections  to  h( 


RESISTIXG  MEDIUM  IX  SPACE. 


557 


applied  to  the  olenients.  For  tliis  purpose  the  partial  dift'erential  co- 
efficients of  the  geocentric  sj)herical  co-ordinates  with  respect  to  x 
and  7/  must  be  determined.  Thus,  if  we  substitute  the  values  of  ft, 
(f,  and  Jf  given  by  (272)  and  (27;3)  in  the  equations  (12).j  and  (14)^,, 
we  obtain 


-J-  =  a  tan  ^  sui  v  — ^  -' -,3— 

dx  2t  d/j. 


t-T„ 


's, 


dv a'  cos  <p     (t  —  Tj) 

^  —  —  7  2r 


dr  t  —  T„     ,_, 

- ,-  =  —  a  cos  <p  cos  V ,    (2(4) 

dy  r 


dv      I      2       ,  ^  \  .      t-T„ 

=1 -u  tan  V  cos  v   sm  v ^> 

cty      \  cos  ^  /  T 


dy 

in  which  s  =  20G2G4".8,  fi  being  expressed  in  seconds  of  arc.  Com- 
binintr  the  results  thus  obtained  with  the  differential  coefficients  of 
the  geocentric  spherical  co-ordinates  with  respect  to  /•  and  r,  as  indi- 
cated by  the  equations  (42).2,  we  obtain  the  recpiired  coeiHcients  of  x 
and  y  to  be  introduced  into  the  equations  of  condition.  The  solution 
of  all  the  equations  of  condition  by  the  method  of  least  squares  will 
then  furnish  the  most  probable  values  of  >/  and  x,  or  of  the  secular 
variations  of  the  eccentricity  and  mean  motion,  without  any  assump- 
tion being  made  in  reference  either  to  the  density  of  the  ethereal  fluid 
or  to  the  modifications  of  the  resistance  on  account  of  the  changes  in 
the  form  and  dimensions  of  the  comet,  and  the  results  thus  derived 
may  be  employed  in  determining  the  values  of  3T,  fi,  and  (S  for  the 
subsequent  returns  of  the  comet  to  the  perihelion. 

In  all  the  cases  in  which  the  periodic  comets  have  been  observed 
sufficiently,  the  existence  of  these  secular  changes  of  the  elements 
seems  to  be  well  established;  and  if  we  grant  that  they  arise  from  the 
resistance  of  an  ethereal  fluid,  the  total  obliteration  of  our  solar 
system  is  to  be  the  final  result.  The  fact  that  no  such  inequalities 
have  yet  been  detc>cted  in  the  case  of  the  motion  of  any  of  the  planets, 
shows  simply  the  immensity  of  the  period  which  must  elapse  before 
tlie  final  catastrophe,  and  does  not  render  it  any  the  less  certain. 
Huch,  indeed,  appear  to  be  the  present  indications  of  science  in  re- 
gard to  this  important  question ;  but  it  is  by  no  means  impossible 
that,  as  in  at  least  one  similar  case  already,  the  operation  of  the 
simple  and  unique  law  of  gravitation  will  alone  completely  explain 
these  inequalities,  and  assign  a  limit  which  they  can  never  pass,  and 
thus  afford  a  sublime  proof  of  the  provident  care  of  the  Omxii'otkxt 

CUE.VTOK. 


TABLES. 


659 


18 
HI 
20 

'^1  I 
•i-i  I 
23     ( 


30 


33 


34 


35    0 


36 


TABLE  I.    Angle  of  the  Vertical  and  Logarithm  of  the  Earth's  Radius. 

1 


Arijiiriii'iit  0 -t:=  f!f'Ofrr:i|iliiciil  I,;iiitii(li'. 


{ 'iiiiiiirts>mii 


l!!t',l.l5 


1) 

(1 

1 

0 

•i 

II 

:< 

II 

1 

(1 

5 

0 

(i 

0 

4 

II 

H 

II 

» 

II 

10 

(1 

11 

II 

\i 

II 

\.i 

II 

11 

II 

15 

0 

l<> 

0 

17 

0 

IH 

0 

1!) 

0 

20 

n 

•il 

0 

•i'i 

II 

33 

0 

21 

0 

a.) 

0 

'H\ 

0 

27 

II 

28 

(1 

'id 

0 

30 

(1 

III 

L'O 

:;i) 

■lU 

aO 

31 

0 

10 

L'O 

:;(! 

40 

50 

32 

II 

111 

20 

:iii 

■(0 

.00 

33 

(1 

10 

20 

:0 

40 

oO 

U 

0 

10 

20 

;)o 

40 

50 

35 

0 

0-*' 


9 
9 

10 


lO 


10 


10 


10 


o.oo 
24.02 

4S.02 
I195 

35. So 
59-54 
23.12 
46.54 
9.76 
32.74; 

5547! 
17.921 

4  40.06 

5  ■•S5 
5   ^V^^, 

5  44-33 

6  4.95: 
6  25.14, 

44.861 
4.09 
22.80 
40.99 
58.61 
15.66 

32.10 

47-93 
3.12 
17.65 
31.50 
44.66 1 

57-»ij 
59.12! 
I.I  I 
3.07 
5.02 
6.94 

8.85 

IO-73 
12.59 

»4-44 
16.26 
18.06 

19.84 
21.60 

*3-34 
25.05 
26.75 
Z8.43 

30.08 
31.71 
33.32 
34.91 
36.48 
38.03 

39-55 
,}.i.o6 
42.54 
44.00 

45-44 
46.86 

10  48.25 


iiiir. 


24.02 

24.00 

13-93 

^3-^5 

23-74 
23.5S 

23.42 
23.22 
22.98 
22.73 
22.45 
22.14 

21.79 
21.43 
21.05 
20.62 
20. 1 9 
19.72 

9.23 
8.7, 
8.19 
7.62 
7-05 
6-44 
5.83 
5.19 

4-53 
3.85 
3.16 
2.46 

2.00 

1-99 
1.96 

1.95 
1.92 
1. 9 1 

1.88 
1.86 
1.85 
1.82 

1.80 
..78 

1.76 

'•74 
1.71 

1.70 
1.68 
1.65 

1.63 
1.6 1 

'-59 

'-57 
1-55 
1.52 

1. 51 
1.48 
1.46 
1.44 
1.42 
'•39 


h'lrp 


Dill. 


9.999 


9.999 


0.000  0000 
9.999   9996 
99S2i 
996 1 

9930, 
9891! 

9X4  3 1 
9786' 
9721. 
9648 1 
9^661 
9476 

9377 
9-7': 
9'57 
9°35; 
89051 
8768; 

8624I 
8472! 
8314: 
8149 

7977 
7799 
7614 

7424I 
7228: 
7027 
6820' 
6608  i 

9-999  6392' 
6355 
6319; 
6287. 

6245: 
6208 


9.999 


9.999 


9.999 


9.999 


61 7 1 
6134 
6096 
6059 
6021 
5984 

5946 
5908 

5X70 
5«3i 
5794 
5755 

5717 
56-8 

5f .  - 
5(01 

55"- 
55'-' 
S4f.^ 
54--J 
54<'6 
5317 

5327 
5288 

9-999  5*48 


9-999 


9-999 


4 
'4 
21 

3' 
39 
48 

57 
<'5 

73 
82 
90 
99 
106 

"4 
122 
130 

'37 
'44 

152 

158 
165 
172 
178 
185 

190 
196 
201 
207 
212 
216 

37 
36 

37 
>7 
37 
37 

37 
3** 
37 
3« 
37 
38 

3S 
3X 
38 
38 
39 
38 

39 

;8 

:<') 

3S 
39 

39 
39 
39 

40 

39 
40 


39     0 

Ml 

I'll 
.':o 
III 
r.ii 

:iG   11 

10 
20 

,'SII 

1(1 

,00 

37  0 

10 

20 

;io 

40 
611 

38  0 

10 
20 

:!ii 
III 

50 

39  (I 

10 
20 

;io 

40 

50 

40  II 

III 

20 
30 
40 
50 

41  0 

111 

20 

;io 

40 
50 

42  0 

10 
20 
80 
40 
50 

43  0 

10 
20 

;io 

40 
50 

44  0 

10 

20 
:!0 

40 
50 

45  0 


-/>-0' 


10 


10 
1 1 


II 


II 


II 


48.25' 

49.63 

50.98 

52.31 

53.62 

54.90 

56.16 
57-4', 
58-63; 
59-821 

I.OOj 

2.15 

3.28  i 
4-39| 
5-47 
6.54! 
7.58 

8-59: 

9-59' 
10.56 

n.51 

12.44 

'3-34 
14.22 

15.08 
15.92 
16.73 
17.52 
18.29 
19.04 

19.76 
20.46 
21.13 

21.79 
22.42 
23.02 

23.61 
24.17 

24.70 
25.22 

25-7' 
26.18 

26.62 

27.04 
27.44 
27.S2 
28.17 
28.50 

28.80 
29.08 

29-34 
29.58 
29.79 
29.98 

30.14 
30.29 
30.41 

30.50 
30.57: 
30.62 

3°-65 


Hilt. 


1. 38 
'-35 
'•33 
'•)' 
1.28 
1.26 

1.25 
1.22 
1. 19 
1. 18 
1. 15 
'•'3 

I.I  I 

1.08 
1.07 
1.04 
1. 01 

1. 00 

0.97 
0.95 
0.93 
0.90 
0.88 
0.86 

0.84 
o.Si 
0.79 
0.77 
0.75 
0.72 

0.70 
0.67 
0.66 
0.63 
0.60 
C.59 

0.56 
0.53 
0.52 
0.49 
0.47 
0.44 

0.42 
0.40 
0.38 
0.35 

0-33 
0.30 

0.28 
0.26 
0.24 
0.21 
0.19 
0.16 

0.15 
0.12 
0.09 

0.07 
0.05 
0.03 


lot^p 


9-999 


9.999 


9.999 


9.999 


9.999 


5248 
5208 
5169 
5129 
5081) 
5049 

5009 
4909 
4929 
488S 
4848 
48071 

4767; 

4726 

4686I 

4645 

4604' 

4563 

4522! 
4481 
4440 1 

4399 
4358 

43'7| 
4276! 
4234! 
4' 93 
4152 
4110 
4069! 
i 
4027! 

3985 
39441 
3902. 
3860 
3819 

9-999  3777, 
3735: 
3693' 
36511 
3609' 
35671 

9-999  3525 
3483 
3441 1 
33991 

3357! 
33'5j 

3273' 
3230I 
3188I 

3'46' 

3104: 
3062 

3019 

2977' 
2935^ 
28921 

28501 
2808! 

9.999  2766! 


9.999 


9-999 


9.999 


DIfr. 


40 

39 

40 
40 
40 
40 

40 
40 

4' 

40 

4> 
40 

4' 
40 

4« 
4« 
4' 
41 

4« 
41 

4' 

4' 
41 
4« 
42 
4' 
4' 
42 
4' 
42 

42 
4' 
42 
42 
41 
42 

42 
42 
42 
42 
42 
4*. 

4» 
42 

42 
42 
42 
42 

43 
42 
42 
42 
42 
43 
42 
42 

43 
42 

42 
42 


661 


IMAGE  EVALUATION 
TEST  TARGET  (MT-S) 


fe 


.^/ 


■■'■\^   ^. 


«v.    'ii.' 


V 


% 


/, 


^ 


= 

y= 

1.25 


'•      110 


I 


ITS 
22 

2.0 


U    11 1.6 


i^ 


/a 


Vl 


w 


Hiotographic 

Sciences 
Corporation 


as  t'/ESV  MAIN  STREIT 

WEBSTER,  N.Y.  145S0 

(716)  S73-4S03 


V 


iV 


:1>' 


;\ 


\ 


^ 
^ 


[V 


6^ 


<6^ 


TABLE  I.    Angle  of  the  Vertical  and  Logarithm  of  the  Earth's  Radius. 


e'--:r,  (iiHMH'iitrir  Latitixlo. 


/)    -  Kurth'H  Itiuliud. 


r 

J '' 


t 

J. 

I 


i 


* 

1 

♦  -♦' 

Uiir. 

l<'«P 

iMir. 

« 

♦  -*' 

j)iir. 

tf 

1     i-l" 

l"ttp 

iiifr. 

o          1 

1            11) 

'            ft 

II   30.6? 

If 

0.00 

9.999  2766 
2723 

i 
43 

0 

55 

II 
III 

10  49.7A 
48.36 

9-999  0*75 
0235 

40 

2(1 

0.01 

0.05 

2681 
2639 

4» 
41 

20 

.'ill 

46.97 

45-51 

,      '-39 
1.42 

0195 
0155 

4^ 

(       4' 

ID 

3o.,-i 
30.42 

0.07 

0.09 

'     0.1 1 

2596 
»S54 

43 
4» 
41 

III 
50 

44.11 
42.65 

1.4A 
1.46 
1.49 

0116 

0076 

!       39 

4' 
i       3') 

40    II 

11    30.31 

0.4 

0.10 

9.999  2512 

!      4* 

43 
,     42 
42 
43 
4» 

50 

0 

10  41.16 

9.999  0037 

III 

30.17 

2470 

III 

39.65 

1. 51 

9.998  9998 

39 

'20 

;i(i 
III 

30.01 
li;.tf2 
iy.61 

0.19 

0.21 

0.2, 

1    0.26 

2427 
13X5 
»143 

211 
.'111 

to 

38.11 
36.5^ 

35.01 

1. 52 
'•55 

'•57 

9958 

99 '9 

9880 

40 
39 

11' 

6(1 

29. 3  X 

2300 

40 

334' 

!•"  0 

1.61 

9841 

39 
39 

'     47    0 

II     2C).I2 

9.999  2258 

57 

0 

10   31.80 

1.64 

1.66 
1.67 

9.998  9802 

111 

211 

iX.S? 

28.54 

0.27 
0.31 

2216 
2174 

4* 
4» 

10 

20 

30.16 
28.50 

9764 
97*5 

31* 
39 

.■(0 

i8.ia 

o*3<t 

2132 

4* 

.'ill 

26.83 

9686 

39 

411 
.-.0 

17.87 
27.50 

0.35 
0.37 
0.40 

2089 
2047 

43 
4S 
4* 

10 

50 

25.13 

23.40 

1.70 
'-73 

'■74 

9648 
9610 

3" 
39 

48    II 

II   27.10 

9.999    2005 

68 

0 

10  21.66 

1.76 

1.79 
1.80 
1.83 
1.85 
1.86 

9.998  9571 

10 

1            20 
.'10 
40 

2(1.69 
26.24 
25. 7X 
25.29 

0.41 
0.45 
0.46 
0.49 
0.51 
0.54 

1963 
1921 

1879 
1837 

41 
41 

4S 
4» 

10 
211 

:iii 
10 

19.90 
18  II 
16.31 
14.48 

9533 
949  < 
9457 
94'9 

3" 
3« 
3« 
3S 

50 

44-7« 

"795 

.'ill 

12.63 

938* 

15 

'40    II 

II    14.24 

0.55 
0.58 
0.61 
0.63 
0.65 
0.67 

9999  "753 

4» 

5» 

0 

10   10.77 

1.89 

9.998  9344 

1" 

23.69 

1711 

10 

8.88 

9307 

37 

SO 

23... 

1669 

4* 
41 

20 

6.97 

1. 91 

9269 

31* 

80 

22.50 

1627 

.•50 

3.08 

'-93 
1  96 

9232 

37 

40 

21.87 

15S6 

4' 

40 

9'95 

3' 

1            90 

21.22 

•544 

4* 
4» 

50 

10     I. II 

1.97 
1.99 

9158 

37 
37 

5U    0 

1            I*) 
30 

II    20.55 
I9.S5 
19.13 

0.70 
0.72 

9.999  1502 
1460 
1419 

'377 

'335 
1294 

4i 
4« 

00 
01 
0-.J 

0 
0 
0 

9  59'» 
9  46-74 
9   33-65 
9   19.85 

9     5-36 
8   50.21 

12.38 
1 3.09 

9.998  9121 
8902 
86S8 

219 

-'4 

80 
40 

'             SO 

18.39 

17.63 
16.84 

0.74 
0.76 

0.79 

0.82 

42 

i' 

4- 

0:1 
04 
05 

0 
0 
0 

13.80 
14.49 
15.15 

15.81 

8479 
8275 

8077 

109 

Icjil 

'91 

,      51     ii 

II     16.02 

0.83 
0.86 

o.Sg 
0.90 
0.93 
0.95 

9.999   'i5» 

4' 
4« 
41 
4« 
4' 
4« 

00 

0 

8    34.40 

16.43 

17.05 
17.63 
38.21 
18.75 
19.27 

9.998   7884 

187 
iSo 
'7; 
I6!l 

161 
'54 

10 
SO 
80 
40 

:,o 

15.19 

«4-33 
»3-45 
12.55 
11.62 

1211 

1170 
1128 
1087 
1 046 

07 
OH 
00 
7« 
71 

0 
0 
0 
0 
0 

8    17.97 
8     0.92 
7  43-29 
7   i5-ol*l 
7     6.33| 

7697 
75'7 
734* 
7174 
7013 

■    5a   (1 

II    10,67 

9.999   1005 

72 

0 

6  47.06 

19.78 
20.25 

9.998  6859 

146 
140 

10 

i             20 

i;° 

0.97 
0.99 

0963 
0922 

4- 
4« 

7;i 

74 

fl 

0 

6  27.28; 
6     7.03I 

6713 
6573 

80 

7.69 

1.02 

088' 

4' 

75 

0 

5  4633 

20.70 

6441 

132 

1            40 

r.n 

6.66 
5.60 

1.03 
1.06 
1.09 

o84„ 
0800 

4' 

40 

4' 

70 
77 

II 

0 

5  25.20 
5     3-67 

21.13 
21.53 
21.90 

63.7 

02011 

I 

'*4 
116 

101! 

5:}    (1 

III 

II     4.51 
3.40 

I.IX 

9.999  0759 
0718 

4' 

78 
70 

0 
0 

4  4'-77: 
4    "9-53 
3   56-96, 

22.24 

9.998  60931 
5993 

100 

1          -" 

4.27 

I.I3 

0677 

4' 

8<) 

0 

22.57 

22.86 

23.12 

*3-35 
23.56 

5901: 

I] 

75 

6- 

57 

:io 
III 
."io 

II      1.12 

10  59.94 

58.74 

1.15 

1. 18 

1.20 
1.22 

0637 
0596 
0556 

40 

4' 

40 

4' 

81 

82 
8U 

0 
0 
0 

■*      -^                 1 

3  34-'o, 
3    'o-9«' 
2  47.63 

1 

5818! 

54     0 

10 

1 

10  57.52 
56.28 
55.02 
53-73 

I.2A 
1.26 
1.29 

9.999  0515 
0475 
043s 
0395 

4C 
40 

40 

81 
85 
80 

87 

0 
0 

f 

0 

2  24.07 
2     0.33 
I    36.44 
I    12.43 

»3-74 
23.89 
24.01 

9.998  5619 

5570 
553° 
549.'< 

49 

40 

3» 
1: 

'3 

5 

» 

52.42 

I. 31 

0355 

40 

88 

0 

0  48.3A 
0  24.18 

24.0., 
24.16 
24.18 

5476 

•f 

51.09 

'•33 
"•35 

0315 

40 
40 

80 

0 

S4O3 

55     0 

10  49.74 

9.999  0275 

00 

0 

0     0.00 

9.998  5458 



662 


irth's  Radius. 


TABLE  n. 

I'lir  converting  interval)*  of  Mean  Solnr  Tinie  into  ii|iiival<.>nt  infervniM  :;f  Sidcrea!  Time. 


I«KP 


urn. 


9.998 


,999  027  V 

023s 
0195 

o»55 
0116 

0076 

.999  o°?7 
1.998  9998 

995*1 

9880I 

9841 1 

^.998  9802 
9764 

91*5! 
96861 

9648  i 
96101 

957« 
9533 
9495 
9457 
9419 

938*1 

9.998  9344, 

9307; 
9169I 

9»3»1 
9>95i 
91581 

9.998  91 21 1 

8902 
8688 
8479 

8275 
8077, 

9.998   7884' 
7697 

75«7 
734»| 
7174 
7013 

9.998  6859' 
6713 

6573 
6441 
6317 
5201 

9.998  6093, 

5993 
5901  i 
5818! 

5676^ 
9,998  5619 

557° 
553° 
549^ 
5476 
54<^'J, 
9.998  5458 


40 
4^ 
4= 
39 
45 
3') 

V) 
40 

3') 
3'* 
3') 
V) 

3^ 
3') 
3') 
3* 
3^ 
39 

3'^ 
3" 
3* 
3!< 
V 
3^ 

37 
3* 
37 
37 
37 
37 

211) 

J14 

loq 
204 
198 

'93 

18: 

iSo 

\bl 
161 
'54 

146 
140 
'3- 

116 


100 

f, 

75 
6' 

57 

49 

40 

3» 
11 

'3 

5 


lluun. 


M'-nii  T.     i>l<lrr<>al  Tiiii<-. 


h 
I 

2 

3 
4 


7 
8 

9 
10 
1 1 
12 

"3 
'4 
'5 
16 

>7 
18 

>9 

20 
21 
22 
»3 
»4 


7 
8 

9 
10 
II 
12 

«3 
'4 
«5 

16 

•7 
18 

>9 

20 
21 

22 

13 
»4 


9.856 
19.713 
29.569 
39.426 
49.282 
59«39 

8.995 
18.852 
28.708 

38.5f'5 
48.421 
58.278 

8134 
17.991 

27.847 

37-704 
47.560 
57.416 

7-»73 
17.129 
26.986 
36.842 
46.699 
56-555 


.S   r. 


'J-    St 

S  .5   o 

•25  = 

^    i   I 

S  -   g 

<.  ti 

o  .S    CI 

J -it 

4<     V     — 


£ 


r  S  5 


I  a 

^  1 
IS  » 


+ 


MiMiii  T. 


2 
3 
4 
5 
6 

7 
8 

9 
10 
II 

12 

«3 
'4 
>5 

16 

17 
18 

>9 

20 
21 

22 

23 
»4 

II 

17 
28 
29 
30 

3' 

3» 
33 
34 

II 

]l 

39 

40 

41 

4» 

43 

44 

46 
47 
48 

49 

50 
5« 

;a 

53 
54 

II 
H 
11 


Miiiutt>ii. 

»< 

»-uuibi. 

UvciuiaU. 

8lderml  Time 

Uiaii  T. 

8iil(>rpiil  TInio 

Mean  T. 

8(<li-rral  Time. 

m        I 

1 

( 

( 

< 

1       I   0.164 

1 

1.003 

0.02 

0.020 

1       2  0.329 

2 

2.005 

O.OA 
0.00 

0.040 
0.060 

3  0-493 

3 

3,008 

4  o-<'57 

4 

4,011 

0.08 

o.c8o 

5  0.811 

6  0.986 

5 

6,016 

0. 10 

0.100 

6 

0.12 

0.120 

7  >'5o 

8  1.34 

9  '-478 

7 
8 

7,019 
8.022 

O.IA 
0.16 

O.IAO 

0.160 

9 

9,025 

0.18 

o.)8o 

10  1.643 

10 

10.027 

0,20 

0  20| 

II    1.807 

II 

11.030 

0.22 

0.221 

11  1.971 

12 

12.033 

0.24 

0.241 

13  2.136 

"3 

13.036 

0.26 

0  261 

14  2.300 

'4 

14-038 

0.18 

0.281 

1     15   1-46-4 

I     16  2.628 

»5 

15.041 
16.044 

0.30 

0.301 

16 

0.32 

0.321 

17   2,793 

«7 

17.047 
18.049 

O.3A 
0.36 

0-3.41 
0.361 

18  2.957 

18 

19  3.121 

>9 

19.052 

0.38 

0.381 

!     20  3.285 

20 

20.055 

0.40 

0.401 

,     »«   3-45° 

21 

21.057 

0.42 

0.421 

,     »»  3-6I4 

22 

22.060 

0-44 
0.46 

O.4AI 
0.461 

i     13  3-778 

13 

23.063 
24.066 

H  3-943 

14 

0.48 

0.481 

25  4.107 

25 

25.06' 
26.071 

0.50 

0.501 

1     26  4-271 

26 

0.52 

0.521 

17  4-435 
1     28  4  600 

17 

27.074 

0.5  A 
0.56 

0-541 
0.562 

28 

28.077 

I     29  4-76.^ 
30  4.928 

19 

29.079 

0.58 

0.582 
0.602 

30 

30.082 

0.60 

31   5x92 

3' 

31.085 

0.62 

0.622 

i     31  5-157 

31 

32.088 

0.64 

o.6a2 
0.662 

1     33  5.421 

33 

33.090 

0.66 

1     34  5-585 

34 

34-093 

0.68 

0.682 

35  5-750 

35 

35.096 

0.70 

C.702 

36  5.914 

36 

36.099 

0.72 

0.722 

37  6.078 

37 

37,101 

0-74 
0.76 

0-741 

0.762 

;    38  6,242 

38 

38.104 

39  6.407 

39 

39.107 

0.78 

0.782 

40  6.571 

40 

40. 1 1 0 

0.80 

0.802 

41  6.735 

4' 

41.112 

0.82 

0,822 

42  6,899 

41 

42.115 

0.84 

0.842 

43  7-064 

t3 

43.118 

086 

O.S62 

,     44  7,228 

44 

44,120 

0.88 

0.882 

;     45  7-39* 
:     46  7-557 

Jl 

45-'i3 
46.126 

0.90 
0.92 

0.902 

0.923 

47  7-71' 

47 

47.129 

0.94 

o.v^3 
0.963 

i     48  7-885 

48 

48.131 

0.96 

i     49  8.049 

49 

49 '34 

0.98 

0.983 

1     50  8.214 
;     5'   8.378 

50 

50.137 

1. 00 

1.003 

5> 

51.140 

52  8.542 

51 

52.142 

1     53  8-707 

53 

53-145 
54.148 

1     54  8.871 

t 

54 

;      55    9-035 

55 

S5»5' 
56.153 

S7-'50 

1     56  9  "99 

56 

57  9-364 

58  9.52S 

57 

58 

58-159 

59  9-691 

60  9.856 

f,2 

59.162 
60.164 

jua 


TABLE  m. 

For  convertinf{  intcrvalfi  of  Sulcrcal  Tiiiu!  into  «-(iuivalcnt  intervals  of  Mean  Solar  Time. 


1 

— - 

■ 

-    - 

m 



■' 

■ 

Hour 

Iloure. 

Miiiutva.             1 

Socdiiila.               1 

DcciuiaU. 

1 

I 
>> 

4 

j    81.1.  T. 

Mean  Time, 

«ia.  T. 

Mmiii  TIiiip. 

Sid.  T. 

Mean  Tiino. 

Sl.l.  T. 
i 

Meuu  Time. 

h 

Am          1 

m 

m         1 

« 

f 

( 

I 

0   59   50.170 

I 

0   59.836 

I 

0-997 

0.02 

0.020 

H 

•> 

2 

I    59   40.341 

2 

I    59.672 

2 

1.995 

0.04 
0.06 

O.OAO 
0.000 

H 

0 

3 

2  59   30.511 

3 

2     59.509 

3 

2.991 

^1 

4 

3   59  20.6S1 

4 

3   59-345 

4 

3.989 

0.08 

0.080 

H 

7 

1 

4  59  lo-^S* 

1 

4  59  «X« 

I 

4.9X6 

0.10 

0.100 

H 

K 

5  59     1.023 

5   59.017 

5.9X4 

0.1 2 

0.120 

■ 

i) 

7 

6  58  51.193 

7 

6  58.853 

7 

6.981 

0.14 
0.16 

0.140 

I 

10 
11 

8 

7  58  4«-3<'3 

? 

7  58.6X9 

8 

7-978 

0.160 

H 

9 

«  S«  3<-534 

9 

8   58.516 

9 

8.975 

0.18 

0.180         i 

H 

12 

lO 

9  5*^   2«-704 

10 

9   5X362 

10 

9-973 

0.20 

0.199 

^1 

1:) 

II 

10  58   11.875 

II 

10  5X.198 

II 

10.970 

0.22 

0.219 

^^ 

12 

II   58     2.045 

12 

II    58.034 

12 

11.967 

0.24 

0.239 

H 

14 
l.'> 
IK 
17 
IS 

«3 

12  57   52.216 

«3 

12   57.X70 

•3 

12.964 

0.26 

0.259    ! 

H 

14 

13  57  42.3X6 

»4 

13   57.706 

>4 

13.962 

0.28 

0.279 

H 

•S 

'4  57  3»-557 

;^ 

«4   57543 

'5 

14.959 

0.30 

0.299 

^1 

i6 

15  57  22.727 

«5    57-379 

16 

15-956 

0.32 

0.319 

^^ 

>7 

16  5-'   12.X97 

17 

16  57.215 

'Z 

16.954 

O.3A 
0.36 

0-339 

^^ 

li» 
20 
21 

l8 

17  57     3.068 

18 

17  57-051 

18 

17.951 

0.359 

H 

>9 

18   c6  53.238 

»9 

18  56.887 

'9 

1X.948 

0.38 

0.379 

^^1 

20 

19  56  43.409 

20 

19  56.723 

20 

19-945 

0.40 

0.399 

^1 

22 

21 

20  56  33-579 

21 

20  56.560 

21 

20.943 

0.4.2 

0.419    ; 

^1 

•»:i 

22 

21     56    ■13  750 

2^ 

21   56.396 

22 

21.940 

0-44 
0.46 

0.439 

^1 

24 

13 

22    56     13.910 

»3 

22  56.232 

23  56.068 

24  55.904 

*3 

22.937 

0.459 

^1 

24 

23    56       4.091 

*4 

*4 

11 

23.934 
24.932 

0.48 
0.50 

0.479 
0.499 

I 

26 

25   55.740 

25.929 

0.52 

0.519 

^H 

ti 

2*§ 

*6  55-577 

*7 

16.926 

0.5A 
0.56 

0.539 

^1 

»7   55-4' 3 

28 

27.924 

0.558 

^H 

\ 

-t     g 

19 

28  55.2A9 

29  55.0X5 

30  54.911 

29 

iX.921 

o.<8 
0.60 

0.62 

0.578 

^1 

1 

2  i 

30 

3» 

30 
3« 

29.918 
30.915 

0.59X 
0.618 

■u 

B  "2 

3» 

31   54.758 

32 

31.913 

0.64 
0.66 

0.638 

^1 

3  'S     , 

33 

32  54.594 

33 

32.910 

0.65X 

^1 

t 

1  =?a 

34 

33  54-430 

34 

33-907 

0.68 

0.67X 

^H 

2l2 

5^5 

1^ 

34  S4-if'6 

35 

34.904 

0.70 

0.69X 

^H 

35   S4-«02 

36 

35.902 

0.72 

0.71X 

^1 

37 

36  53.938 

37 

36.899 

0-74 
0.76 

0.7  3X 

H 

§  « :? 

3« 

37   53-775 

3X 

37.X96 

0.758 

^H 

S   J3     X 

a  —   fi 
^   rt   ?i 

39 

38   53.611 

39 

3X.X94 

0.78 

0.778 

^1 

i 

40 

39  53  447 

40  53.283 

40 

39-S2i 

0.80 

0.798 

^H 

< 

M? 

4> 

4« 

40.X88 

0.82 

0.818 

^H 

4a 

41    53.119 

42 

41.885 

0.84 

0.83X 

^1 

i  =  - 

43 

42   52.955 

43 

42.883 

086 

0.858 

^1 

£  g  2 

44 

43   5*-79» 

44 

43.8X0 

0.88 

0.87X 

^1 

&  a  s 

45 

44  52.628 

45 

44.X77 

0.90 

0.89X 

^H 

1   il-3 

46 

45   5»-464 

46 

45-874 

0.92 

0.917 

^H 

47 

46  52.300 

47 

46.872 

0.94 

0.937 

^H 

s|i 

48 

47  5»->36 

48 

47.869 

0.96 

0.957 

H 

b   s   S 

49 

48  51.972 

49  S'-«09 

49 

48.866 

0.98 

0.977 

^1 

t 

SO 

50 

49.863 

1. 00 

0.997 

^H 

I 

S  «  T 

S> 

50  51.645 

SI 

50.861 

^H 

s  a  + 

Sa 

51   51.481 

s» 

51.858 

1 

^H 

s  -^ 

53 

5»  5«-3>7 

53 

52.855 

^H 

s  a 

54 

53  S"-iS3 

54 

53-853 

^1 

2  ■ 

11 

54  50990 

\l 

54.850 

^1 

S  a 

m    S 

55  50.826 

56  50.662 

55-847 

^H 

li 

57 

56.844 

^H 

i 

5S 

57  50-498 

58 

57.842 

^H 

II 

58  50.334 

it 

58-839 

^H 

59  50>7o 

59.836 

I 

604 


can  Solar  Time. 


TABLE  IV. 

For  convortinR  Ilourst,  Miniite.'«,  and  Sccondft  into  Dcoimals  of  a  Day. 


Deciinali. 

- 

.Menu  Time. 

,       ! 

O.OIO 

O.OJ.0 

o.ooo 

0.080 

O.I  00 

0.120 

0.140 

0.160 

0.180      ' 

0.199 

0.119 

0.239 

0.2S9    : 

0.279 

0.299 

0.319 

0.339 

0.359 

0.379 

0.399 

0.419     1 

0.439 

0.459 

0.479 

0.499 

0.519 

0.539 

0.558 

0.578 

0.598 

0.618 

0.638 

0.658 

0.678 

0.698 

0.718 

0.738 

0.758 

0.778 

0.798 

0.818 

0.838 

0.858 

0.878 

0.898 

0.917 

0.937    ' 

0.957 

0.977 

0.997 

1 

lloiirn. 

1 

Decimal. 

.Min. 

Dttiuinl. 

Mill. 

81 

Dtcimal. 

S.T. 

Drciinal. 

8<-c. 

Dccinml.    1 

0.0416  4- 

1 

.000694  4- 

.021527  + 

1 

.00001 16 

31 

.00035S8 

0 

.0833  + 

2 

.001388  -| 

82 

.022222  -)• 

2 

.0000231 

32 

.0003704 

» 

.1250  + 

8 

.002083    f- 

88 

.022916    i 

3 

.0000347 

33 

.0003819 

4 

.1666  4- 

4 

.002777  ■■}- 

84 

.023611    I 

4 

.0000463 

34 

.0003935 

5 

.2083  -i- 

5 

.003472  -;- 

85 

.024305  -f 

5 

.0000579 

85 

.0004051 

« 

.2500  + 

« 

.004166  -] 

80 

.025000  -|- 

0 

.0000694 

84i 

.C004167 

i 

0.2916  -f 

7 

.004861    r 

87 

.025694-:- 

7 

.0000810 

37 

.0004282 

H 

•3333  ^- 

H 

.005555   1- 

8H 

.026388    •- 

H 

.0000925 

38 

.0004398 

» 

.3750-t- 

» 

.006250  1 

80 

.02708^    •• 

0 

.0001042 

30 

.0004514 

U) 

.41 6  J   t- 

10 

.006944  4- 

40 

.02-777-4- 

10 

.0001 1 57 

40 

.0004630 

11 

•45J'3-|- 

11 

.00-658    f 

41 

.028472-1- 

11 

.0001273 

41 

.0004745 

1-2 

.  5000  |- 

12 

•008333 -i- 

42 

.029166   ■ 

12 

.0001389 

42 

.0004861 

18 

0.5416-)- 

18 

.009027  -j- 

48 

.029861  -1 

18 

.0001505 

43 

.0004977 

14 

•5833  + 

14 

.009722  4 

44 

•030555 -r 

14 

.0001620 

44 

.0005093 

1.1 

.6150    ;- 

ir» 

.010416  -j 

45 

.031250  -;- 

15 

.0001736 

45 

.0005208 

in 

.66664- 

l(i 

.OIIIII     4 

40 

•031944  + 

10 

.0001852 

40 

.0005324 

17 

.7083  + 

17 

.011805  • 

47 

.032638  i- 

17 

.0001968 

47 

.0005440 

is 

.7500 -f 

IN 

.012500  -(- 

48 

•033333  + 

18 

.0002083 

48 

.0005556 

1ft 

0.7916  4- 

ID 

.013194-.- 

40 

.034027-;- 

10 

.0002199 

40 

.0005671 

H> 

■8333  t- 

20 

.013X88-:- 

50 

.034-'22-|- 

20 

.000231 5 

50 

.0005787 

21 

.8750  1- 

21 

.014583  1- 

51 

.035416  i- 

21 

.0002431 

51 

.0005903 

4>i 

.9166  + 

22 

•015277  + 

52 

.036111    :  - 

22 

.0002546 

52 

00060 1 9 

2:1 

0.9583  -f- 

2» 

.015972    : 

58 

.036805  -l- 

28 

.0002662 

53 

.00061 34 

24 

I.OOOO  -\- 

24 

.016666-1- 

54 

.037500  i- 

24 

.0002778 

54 

.0006250 

25 

.017361  4- 

55 

.0381944- 

25 

.0002894 

55 

.0006366  1 

2« 

.018055  -f 

50 

.038888  -i 

20 

.0003009 

50 

.0006481 

27 

.018750  -i 

57 

.039583  -J- 

27 

.0003125 

57 

.0006597 

2S 

•019444  + 

5") 

.040277  -f 

28 

.0003241 

58 

.000671  3 

20 

.020138  -1- 

50 

.040972  -\- 

20 

.0003356 

50 

.0006829  ' 

80 

.020833  ■■• 

00 

.041666  i- 

30 

.0003472 

(to 

.0006944 

Thu  sign  -f-,  appondiHl  to  numbiTs  in  tliis  tiible.  Kignillpt  thiit  the  Iu8l  flgur'  ropoatit  to  intinity. 


TABLE  V. 

For  finding  the  number  of  Days  from  the  beginning  of  the  Year. 


Date. 

Com. 

Bia. 

1 

January  0.0 

0 

0 

February  0.0 

3» 

3« 

March  0.0 

59 

60 

April  0.0 

90 

9' 

May  0.0 

120 

12! 

June  0.0 

'5> 

'5» 

July  0.0 

181 

182 

August  0.0 

212 

213 

September  0.0 

»43 

244 

October  0.0 

*73 

274 

November  0.0 

304 

305 

December  0.0 

334 

335 

5tfa 


TABLE  VI. 

For  finding  the  Tnip  Anomaly  or  iV.c  Timt"  from  tin-  IVriholion  in  n  Pnrnbolic  Orbif. 


V, 

0° 

1° 

2° 

3° 

M. 

DIIT.  1". 

u. 

Dlff.  1". 

M. 

Diflr.  I". 

K. 

DIff.  I" 

0' 

o.cooooo 

iXi.Xi 

0.654532 

't't^ 

1.309263 

181.92 

1.964393 

il'i.05 

1 

0  oio<;oS 

iXi  Xi 

0.665442  , 

1X1.83 

1.32017X 

1X1.92 

1.975316 

1X2.06 

'i 

o  021817 

iXi.Xi 

0.676352 

181.83 

1.331093 

181.92 

1.9X62^0 
1.997164 

1S2.06 

:i 

0.03271s 

iXi.Xl 

0.6X7262 

1X1X4 

1.34200X 

181.92 

1X2.06 

* 

0.043633 

181.81 

0.69X172 

181.84 

i.35»9»3 

181.92 

1.008087 

182.07 

5 

0.05454a 

IXI.8I 

0.7090X2 

181.84 

1.363X39 

181.93 

1.019011 

1X2.07 

0 

0.0(15450 

iXi.Xi 

0.719993 

1X1. X4 

'•374755 

1X1.93 

1.029916 
2.040X60 

1X2.07 

7 

007635S 

iXi.Xi 

0.730903 

1X1. X4 

1.3X5670 

1X1  93 

iSi.o: 

8 

0.0X7167 

iXi.Xi 

0.741X13 

'S'-^ 

1.396586 

1X1.93 

2.<-5"7S5 

1X2. cS 

0 

o.09«i75 

IXI.8I 

0.752724 

1X1.84 

1.407502 

181.93 

2.C62709 

182.08 

lO 

o".  1 0908  3 

181  Xl 

0.763634 

181.84 

1.418418 

181.94 

2.073634 

1X2.08 

11 

0.1 199V- 

iXi.Xi 

0.77454  > 

1X1. X4 

1.429334 

181.94 

2.084559 

1X2.08 

\'2 

0. 13090U 

iXi.Xi 

0.7X^4^6 

1X1. X4 

1.440251 

18.. 94 

1.0954X5 

1X2. 09 

13 

o.r+ixos 

iXi.Xi 

0.7963(10 

1X1. X5 

1.451167 

1X1.94 

2.106410 

iX2.0C( 

:    >* 

O.I  52717 

181.81 

O.X07277 

1X1.X5 

1.462083 

1X1.94 

2.117335 

1X2.09 

i     15 

0.163625 

181. 81 

C.818188 

181.85 

1.473000 

181.95 

2.128261 

iX:.ic 

i  Vi 

o.i745!4 

iXi.Xi 

0.829099 

181.85 

1.4X3917 

181.95 

1.139187 

iXi.io 

17 

0.1S5442 

iXi.Xi 

O.X40010 

1X1. 85 

1.494X34 

181.95 

1.150114 

1X2.10 

IH 

0.196350 

iXi.Xi 

0.850921 

1X1.X5 

1.505751 

181.95 

1.161040 

iXi.ii 

lU 

0.207259 

IXI.8I 

0.861832 

1X1.85 

1.516668 

1X1.95 

1.17 1966 

iXi.ii 

20 

0.21  Si  67 

iXi.Xi 

0.872743 

181.85 

«-5i75«5 

181.96 

2.182894 

ix2.II 

'Zl 

0.229076 

iXi.Xi 

O.XX3654 

1X1X6 

1.53X503 

181.96 

1.193810 

iS:.i: 

TZ 

0.2399X4 

iXi.Xi 

O.X94566 

1X1.X6 

1.549420 

181.96 

1.204747 

1S21: 

23 

0.250X93 

IXI.8I 

o.90^47X 

181.X6 

1.560338 

1X1.96 

2.215674 

iSi.i: 

'Z4 

0.261801 

181.81 

0.9163X9 

181.86 

1.571256 

181.96 

2.216602 

182.1^ 

25 

0.272710 

iXi.Xi 

0.927301 

181.86 

1.582174 

181.97 

1.2371:29 

1X2.,  3 

2» 

0.2X3619 

1X1. Xl 

0.93X212 

181. 86 

1.593092 

181.97 

1.148457 

.S2.,; 

27 

0.294527 

1X1. Xl 

0.949124 

181.86 

1.604010 

181.97 

2.2593X5 

1S1.14 

28 

0  305436 

1X1.81 

0.960036 

181.86 

1.614928 
i.625i<47 

181.97 

2.270313 

1S2.14 

2U 

0.316345 

iXi.Xi 

0.970948 

181.87 

181.97 

2.181242 

1X2.14 

i    30 

0.327253 

181.81 

0.981860 

181.87 

1.636766 

181.98 

2.191170 

1X1.14 

31 

0.33X162 

1S1.81 

0.992772 

181.87 

1.6476X4 

181.98 

1.303099 

1S2.15 

:« 

0.349071 

iXi.Xi 

1.0036X4 

181.87 

1.65X603 

1X1.98 

1.314018 

1S2.15 

33 

0.3599X0 

181. Xl 

1.014596 

181.87 

1.669522 

181.98 

2.324957 

lX;.i; 

i    a» 

0.370XX8 

1X1.81 

1.025509 

181.87 

1. 680441 

181.99 

1.335X87 

1S2.16 

!    35 

0.381797 

181.81 

1.036421 

181.87 

1.691361 

181.99 

1.346816 

1X1.16 

30 

0.392706 

1X1.81 

•  047334 

1X1.87 

1.702280 

181.99 

1.357746 

iXm6 

37 

0.403615 

1X1.81 

1.05X246 

181. XX 

1.713200 

181.99 

1.36X676 

iXi.i- 

3H 

0.414524 

1X1.82 

1.069159 

181. XX 

1.724120 

iSi.oo 

1.379606 

i8:.r 

i    3U 

0-415433 

1X1.82 

1.0X0072 

181.88 

1.735039 

18a. 00 

1.390536 

1X2.1- 

40 

0.436342 

181.82 

1.0909X5 
1.101898 

181.88 

1.745960 

1S2.00 

1.401467 

iXl.I.*! 

i     41 

0.447251 

181.82 

1X1.88 

1.756XX0 

181.00 

1.412398 

1S2.18 

12 

0.45X160 

181.82 

1.1I281I 

181.89 

1.767X00 

182.01 

1.423329 

lS2.i8 

43 

0.469069 

181.82 

1.123724 

181.89 

1.77X-21 

182.01 

2.434260 

1S2.19 

44 

0.479979 

181.82 

1.134637 

181.89 

i.789('4i 

181.01 

1.445191 

I X  2.19 

45 

0.490X88 

181.82 

i.i4i;55o 

181.89 

1.800562 

182.01 

1.456113 

iX;.i9 

4« 

0.501797 

181.82 

1.156464 

181.89 

1.811483 

182.02 

1.467055 

182.:" 

'     47 

0.512706 

1X1. X2 

i-«67377 

181.89 

1.X22404 

182.02 

1.4779X7 

iX:.2o 

48 

0,523616 

181.82 

1.178291 

181.89 

>->'333i5 

181.01 

1.48X919 

iX;.:o 

4U 

o.S34S»5 

181.82 

1.189205 

181.90 

1.844247 

181.01 

2.499X51 

iXi.:i 

1     50 

0.545435 

181.82 

1.2001 19 

181.90 

1.855168 

182.03 

1.510784 

1X2.21 

51 

0.556344 

181.82 

1.211033 

181.90 

1.866090 

1X2.03 

2.521717 

1X2.22 

;    52 

0.567254 

1X1.82 

1.221947 

181.90 

1. 877012 

1 8 1.04 

2.532650 

1X2.21 

53 

0.578163 

I8I.X3 

1.232861 

181.90 

1.887034 
1.898856 

181.04 

1.543583 

1X2.2: 

1     »•« 

0.589073 

181.83 

«-»43775 

581.91 

181.04 

i-SS45'7 

1X2.23 

55 

0.599983 
0.610892 

181.83 

1.254689 

181.91 

1.909779 

181.04 

2.565450 
2. 5  76  3  84 

182.23 

50 

181.83 

1.265604 
1.276518 

181.91 

1.920701 

181.04 

182.23 

57 

0.621802 

181.83 

181.91 

1.931614 

182.05 

2.587319 

1X2.24 

58 

0.632712 

181.83 

1.28^433 
1.198348 

181.91 

1.942547 

182.05 

2.598253 

1X2.24 

59 

0.643622 

181.83 

181.91 

1.953470 

182.05 

2.609187 

182.24 

!    00 

1 

0.654532 

181.83 

1.309263 

181.92 

1.964393 

181.05 

1.620111 

182.15 

50tf 


iralKilir  Orbit. 


TABLE  VI. 

For  ilnding  the  Tnio  Anomaly  or  tlie  Tinii'  fn>m  the  I'prihi'lion  in  a  I'urabolic  Orhil. 


I>11T.  1" 


ifl 

o<; 

iSj 

c6 

iSi 

06 

iSi 

06 

1X2 

<^7 

1S2.0: 

l!ti 

07 

iSi.o: 

iSi 

cX 

lii 

oX 

182 

cX 

1S2 

oX 

1S2 

09 

1X2 

ov 

|S2.0I> 

iS: 

IC 

1S2 

.10 

1X2 

.10 

iSl 

.11 

1X2 

.11 

IX2.II 

1X2.11 

1X1  12 
IX2.I2 
1X2.13 

1X2. 15 
1X2.I3 
1X114 
1X2.14 
1X2.14 

1X2.14 

1S1.I5 
IS2.I5 
1X1.13 
1X2.16 

1X2.16 
1X2.16 
1X2.17 
1X2.1'' 

182.17 

1X2.1X 
1X2.1X 

1X2. iS 

1X2.19 
1X2.19 

1X2.19 

1X2,:'^ 
1X2.20 
1X2.20 
1X2.21 

1X2.21 
1X2.22 
1X2.22 
1X2.22 
1X2.23 

182.23 
182.23 
1X2.24 
1X1.24 
I  182.14 

1  182.15 


r. 

40 

5° 

6 

70 
M. 

4. 59 19 '7 

M. 

roir.  1". 

M 

3  276651 

WIT.  I". 

1X2.50 

.M. 

3.934181 

Diir.  1". 

1X2.80 

Ditr.  1". 

0 

2.610112 

iSi.15 

183.17 

1 

1.631057 

182.25 

3.2S7602 

1X2.50 

3-94515' 

1X2.81 

4.603907 

1X3.18 

•» 

2.64li>9j 

181.16 

3.29X552 

1X75, 

3.956119 

1X2. 82 

4.614S98 

183. iX 

Ti 

i.652'>l8 

182.26 

3-309503 

1X2.51 

3..;07o88 

1X2. X2 

4.(125X89 

1X3.19 

4 

2.60^86+ 

182.16 

3.320454 

182.51 

3.97X058 

182.83 

4.636880 

183.19 

5 

2.674800 

182.17 

3.331405 

181.51 

3.989018 

182.83 

4.(147872 

183.20 

» 

2.685756 

1S2.17 

3  3+i35<' 

181.53 

3.99999X 

1X2. X4 

4.658864 

1X3.11 

1 

1.696671 

182.27 

3-3533o>* 

1X2.33 

4.01 096  S 

■.X2.S4 

4.669X57 

1X3.11 

H 

2.707609 

181.18 

3.564260 

1X2.54 

4021939 

1X2. X5 

4.6X0X50 

1X3.22 

V 

2.718546 

181.28 

3-375»«i 

1X2.54 

4.032911 

1X2.86 

4.691X43 

1X3.23 

1«> 

2.719483 

181.19 

3.386165 

182.55 

4.043882 

182.86 

4.702X37 

183.24 

II 

1.740410 

181.29 

3.397118 

182.55 

4o;4!<54 

182.X7 

4.713X31 

'!<3--4 

li 

i.75i35« 

182.19 

3.40X071 

1S2.56 

4.065X26 

1X2.X7 

4.7i4-^i<' 

1S3.25 

i:i 

1.76219s 

182.30 

3.419024 

182.56 

4.07(1799 

1X2.88 

4.733821 

1S3.25 

II 

*-775S33 

181.30 

3.429978 

182.57 

4.0X7772 

IX2.X8 

4.74<''*'6 

1X3.26 

15 

2.784171 

1H1.31 

3440931 

182.57 

4-098745 

182.89 

4-757>''i 

1X3.27 

III 

1.795  no 

181.31 

3.451887 

1 82. 58 

4.109718 

1X2.90 

4.76XX09 

1X3.27 

17 

2..><o6o49 

1S2.31 

3.462841 

1X2.5X 

4.120(192 

1X2.90 

4.779X05 

1X3. 2X 

IH 

1.S16988 

182.31 

3-47  3796 

1X2.59 

4.131667 

1X2.91 

4.790X02 

1X3. 28 

lU 

2.817917 

181.31 

3.484751 

181.59 

4.142641 

182.91 

4.X01800 

183.19 

2(1 

2.838867 

181.33 

3.495707 

181.60 

4.153616 

182.92 

4.811797 

183.30 

'^1 

1.849806 

182.33 

3.506663 

182.60 

4.104591 

1X2.93 

4.X13796 

1X3.31 

'ii 

2.860746 

182.33 

3.517619 

182.61 

4-«75  5f'i< 

182.93 

4-^' 34795 

1X3.32 

a:i 

2.871686 

181.34 

3-5J><575 

182.61 

4.18(1544 

182.94 

4-^45794 

1X3.32 

'ii 

2.X82617 

182.34 

3-539531 

182.61 

4.197520 

182.94 

4.850793 

1X3.33 

2.-1 

2.893567 

181.35 

3.550489 

182.62 

4.20S497 

182.95 

4.867793 

1X3,34 

'.:*> 

2.904 ',.''8 

182.35 

3.561447 

182.62 

4.219474 

182.95 

4.87X793 

1X3.34 

27 

2.91544^ 

182. 36 

3.571404 

182.63 

4.230451 

lXj.96 

4.XX9794 

1X3.3, 

28 

2.926391 

181.36 

3.5X3361 

182.63 

4.241429 

1S2.97 

4.900795 

1X3.30 

2U 

s-93733i 

i«i.3''' 

3-594310 

181.64 

4.25240;! 

181.97 

4.911797 

1X3.36 

3() 

2.948274 

182.37 

3.605179 

182.64 

4.263386 

182.98 

4.922799 

183.37 

31 

2.959117 

181.37 

3.616238 

1X2.65 

4.274365 

182.99 

4.933801 

't^-'it 

:i2 

2.9701159 

181.37 

3  627197 

1X2.65 

4.2X5344 

1X2.99 

4-944«'H 

183.38 

:i;t 

2.981102 

182.38 

3.638156 

182.66 

4.296324 

1X3,00 

4.955X07 

•S3.39 

34 

2.992045 

i8i.3« 

3.649116 

181.66 

4.307304 

1X3.00 

4.966811 

183.40 

io 

3.001988 

181.39 

3.660076 

182.67 

4.318284 

183.01 

4.977X15 

183.4, 

36 

3.013931 

182.39 

3.671037 

IX2.6X 

4.329265 

1X3.01 

4.9XX820 

1X3.41 

37 

3.024875 

182.39 

3.6X1997 

182.68 

4.340246 

1X3.02 

4.999S25 

1X3.42 

3N 

3.035819 

182.40 

3.691958 

182.69 

4.351228 

183.03 

5.010X30 

IXJ.43 

M 

3.046763 

182.40 

3.703920 

182.69 

4.362210 

183.03 

5.021836 

1X3.43 

to 

3.057707 

181.41 

3.7i4«fi 

181.70 

4.373191 

183.04 

5.032842 

1X3..44 

U 

3.068652 

1X2.41 

3-7i5«43 

182.70 

4-3X4'75 

1X3.05 

5043^49 

'■^3-45 

42 

3-079597 

182.41 

3.736806 

182.71 

4-395<5^ 

183.05 

5.054X56 

1X3.4O 

13 

3.090541 

182.41 

3-7477''« 

182.71 

4.40(1141 

1X3.06 

5.005X64 

1X3.46 

41 

3.101488 

I8Z.43 

3.751*731 

182.72 

4.417125 

1X3.06 

5.076X72 

1X3.47 

45 

3.111433 

182.43 

3.769694 

182.72 

4.428109 

183.07 

5.0878X0 

,83.48 

4(1 

3-'i3379 

182.44 

3.780658 

182.72 

4.439093 

1X3.0X 

5.09XXX9 

1X3.4X 

47 

3-«343»'; 

182.44 

3.791612 

182.73 

4.45007.S 

183.08 

5.109X9X 

1X3.49 

4N 

3  >4^27a 

182.44 

3.801586 

1S2.74 

4.461064 

183.09 

5.12090X 

1X3-50 

4U 

3.156119 

181,45 

3.81355, 

182.74 

4.472049 

1X3.10 

5.131918 

183.51 

5(» 

3.167166 

181.45 

3.824515 

182.75 

4.483035 

1X3.10 

5.142929 

1X3.51 

51 

3.178113 

181.46 

3.835481 

182.76 

4.494022 

1X3.11 

5..  153940 

1S352 

52 

3.189061 

181.46 

3.8464.46 

182.76 

4.505008 

1X3.12 

5.164951 

1X3.53 

53 

3.200009 

181.47 

3.857412 

182.77 

4-5 '5995 

1X3.12 

5.1759^') 

1X3.54 

54 

3.210957 

182.47 

3.868378 

181.77 

4.5269X3 

1X3,13 

5.186975 

183.54 

55 

3.221905 

182.48 

3-879345 

181.78 

4-53797' 

'^3-'4 

5.197988 

ii<3-55 

50 

3.232854 

181.48 

3.890312 

182.7S 

4-541^959 

183.14 

5.209001 

,83.56 

57 

3.243803 

182.49 

3.901179 

182.79 

4.559948 

1X3.15 

5.220015 

183-57 

5« 

3-2U75» 

182.49 

3.911146 

181.79 

4-570937 

183.15 

5.231029 

183.57 

5» 

3.265702 

182.49 

3.913114 

182.80 

4.581927 

183.16 

5.242044 

,83.58 

60 

1 

3.276651 

182.50 

^.934182 

182.80 

4.592917 

183.17 

5.153059 

183-S9 

467 


TABLE  VI. 

Fur  fiiuUng  llic  True  Aiioinaly  or  tlit-  Time  t'roiii  the  iVrilit  lioii  in  u  Parabolic  Orbit. 


I 


V, 


o 
I 
•i 
.-{ 
1 

it 
0 

T 
N 
0 

lO 
II 
12 
l.'i 
It 

15 
10 
17 
18 
10 

'M 
'Zl 
TZ 
»3 
24 

25 
20 
27 

28 
2U 

30 
31 
32 
33 
34 

35 
30 
37 

38 
30 

40 
41 
42 
43 
44 

45 
40 

47 
48 
40 

50 
51 
52 
53 
54 

55 
SO 
57 
58 
50 

00 


8^ 


M. 

253059 
264075 
275090 
2S6107 
297124 

30X141 

3'y»59 

n^^>77 
341 195 
35i2>4 

3^-3234 
374154 
385*75 
396296 

4073 '7 

418339 
429361 
440384 

451407 
462431 

473455 
484480 

49550s 
506530 

5'7556 

528583 
539610 

550637 
561665 

572693 
583722 

59475* 
605782 
616812 
627X43 

638874 
6499  6 
660938 
671971 
683004 

694038 
705072 
71 6 1 06 

727141 
738177 

749213 
760250 
771287 
781325 
793363 

804401 
815440 
826480 

8375*0 
848561 

859602 
870644 
881686 
892728 
903771 

91481S 


Diir.  1" 


-59 

59 

.60 

.(>! 
.62 

.62 
.63 
.64 
.65 

.66 

.66 
.67 
.68 
.69 
.69 

.70 
•7« 
-72 
■73 
-73 

•74 
■75 
■75 
.76 

-77 
.78 
■79 
■79 
.So 
.81 

.82 
.8, 
.83 
.84 
.85 

.86 
.87 
.87 
.88 
.89 

-90 
.91 
.92 
.92 
•93 

-94 
■95 
.96 
.96 
-97 


98 

-99 

84.00 
84.01 
84.01 

84.02 
84.03 
84.04 
84.05 
84.06 

184.06 


9^ 


.M. 


5.914815 
V925S59 
5.936904 
5-947949 
5-958995 

5.970041 
5.981087 
5.992134 
6.00  5 1 82 
6,014230 

6.025279 
6.036328 

6.047378 
6.058428 
6.069479 

6.080530 
6.091582 
6.102634 
6.113687 
6.124740 

6-135794 
6.146849 

6.157904 
6.168959 
6.180015 

6.191072 
6.202129 
6.213187 
6.224245 
6.235304 

6.246363 
6.257422 
6.268482 
6.279543 
6.290605 

6.301667 
6.31  2729 
6.323792 
6.334855 
6.345919 

6.356984 
''■36X049 
6.379115 
6.390181 
6.401248 

6.412315 
6.423383 

''■43445« 
6.445520 
6.45(1590 

6.467660 
6.47X731 
6.4X9X02 
6.500874 
6.5 1 1946 

6.523019 
6.534092 
6.545166 
6.556241 
6.567316 

6.578391 


10 

ItllT.  1". 

M. 

184.06 
1X4.Q7 
1X4.08 
184.09 
184.10 

6.17839, 
6. 5X9467 
6.600544 
6.61 1622 
6.622700 

184.11 
1X4.11 
184.12 
'84-13 
184-14 

6.633778 
6.644X57 
6.655937 
6.667017 
6.678098 

184.IS 
184.16 
184.17 
184.18 
184.18 

6.6X9179 
6.70026, 
6.711343 
6.722426 
6.733510 

184.19 
184.20 
184.21 
184.22 
184.23 

6.744594 
6.755679 
6.766764 
6.777850 
6.788937 

184.24 
184.25 
184.25 
184.26 
184.27 

6.800024 
6.Xii,,2 
6.822200 
6.833289 
6.844378 

184.28 
184.29 
184.30 
184.31 
184.32 

6.855468 
6.S66559 
6.877650 
6.S88742 
6.X99834 

184.32 
184.33 
184.34 

'84-35 
184.36 

6.910927 
6.922021 
6.933115 
6.9442,0 
6.955305 

184.37 
1X4.38 
184.39 
1 84.40 
184.41 

6.966401 
6.977498 
6.988595 
6.999693 

7.0,079, 

184.4, 
184.42 
,84.43 
184.44 
184.45 

7.021890 
7.032990 
7.044090 
7.055,91 
7.066292 

184.46 
184.47 
184.48 
184.49 
184.50 

7^077394 
7.088497 
7.C99600 
7.110704 
7. ,21808 

184.5, 
184.52 
184.52 

•84-53 
184.54 

7., 32913 
7.144019 

7^'55i*5 
7.166232 
7.177340 

•84-55 
,84.56 
,84.57 
,84.58 
,84.59 

7.188448 

7^'99557 
7.2,o'i66 
7.i2k/76 
7.232886 

184.60 

7.243997 

iMir.  1". 

84. 60 

84. 6 1 
X4.62 
84.63 
84.64 

84-65 
8466 
8467 
84.67 
84.68 

84.69 
84.70 

84-71 
84.72 

8473 

84.74 

84.7s 

4.76 

84-77 
84,78 

84.79 
84.X0 
X4.8, 
84.82 
84.83 

84.84 

84-85 
X4.86 
84.87 
84.88 

84.X9 
X4.90 
X4.91 
84.92 
84.93 

84.94 
84.95 
84.96 
84.97 
84.98 

84.99 
85.00 
85.0, 
85.02 
85.03 

85.04 

85-05 
85.06 

85.07 
85.08 

85.09 
85. ,0 
85.11 
85.12 
85.13 

85.14 
85., 5 
85. 16 
85.17 
85.18 

185.19 


11 


M. 


*43997 
255109 
16622; 

*77335 
288449 

199563 
310678 
321793 
332909 
344026 

355144 
36(1262 

37738" 
3X8500 
399620 

4io74« 
421862 
432983 
444106 
455*30 


354 
8 


Ditr  I" 
85-19 

8v20 
85. 21 
Xv2i 
85.2, 

85-2,- 
Xvlb 

85.27 
X5.2.S 
85.29 

X5.50 
Xi.;. 
85.3* 
85-33 
85-3+ 

85-35 
8^.36 

85-37 
85-38 
85.39 

85-40 
'.5.41 

5-4J 


X; 


.466 

■477^t7 
.488603 

499729 

510X55 

521982 
533110 

544*39 
555368 
5664.97 

5776-8 

588759 

599890 
.6,1022 
,622155 

,  ,633289 

7.644423 

7-655558 
7.666694 
7.677830 

,688967 

7.700,04 
7.7,1242 
7.72238, 
7.733521 

7-74466I- 
7.755802 
7.766943 
7.778085 
7.789228 

800372 
81,516 
822661 
833807 
844953 

856100 
867247 
878396 
889545 
900694 

911845   I  IS5.: 


«5^44 

85.46 
85^47 
85^48 
X5.49 

85-50 


5' 

5i 
53 
54 
55 

85-57 
85.58 
85.59 
85.60 
X5.61 

S5.62 

85-63 
Xv64 
X5.65 
X5.66 

X5.6X 
85.69 

X5.70 
X5.71 
85.72 

85-73 
85-"4 
85^75 
X5.76 
85.78 

X5.79 
8s. Xo 
X5.81 
85. 82 
85.83 


568 


ubolif  Orliii. 


TABLE  VI. 

For  (inilinj;  [\w  True  .Vnoiiiiily  or  ilu-  Tiiiif  tniiii  tlio  Pcrilu'linii  in  m  I'liialnilic  ()il)it. 


11 


\l. 


nvv7  I 

77335 
8)i449 

99  i'' 3 

2  1793 

32' 9°  9 
44026 

55'44 
()(i2(ii 

773«' 
X«500 
(^9620 

.I07ii 
.21862 

.3i9«3 
^44106 

^55^3o 

t66354 

t7747i< 
1.8X603 
^99729 
510X55 

521982 
533110 

44139 

5536S 

,66497 

5776-8 
5X8759 
599890 

II  I022 
^22155 

533289 

S44423 

'SS55« 
366694 
&77830 

588967  ' 

700104 
711242 
722381 

733;*» 

744661- 

755802 

766943 

778085 

789228 

S00372 
511516 
822661 
833807 
S449S3 
S56100 
867247 
S78396 

58954S 
300694 

911845 


DIff.  1" 

185.19 
185.20 
1X5.21 
1X5.21 
185.23 

1X5.25 
1X5  :b 
185. ;7 
lX5.2,S 
1X5.29 

1X5.50 
1S5.51 
1X5.31 

"<v33 
1X5.34 


185.35 

|X5.3(, 

1X5.37 

1X5. 3X 

1X5.39 

1X5.40 

.S5.4' 

1X5.41 

1X5.45 

•'<544 

1X5.46 

1X5.47 

1X5.4X 

1X5.49 

1X5.50 

i85-^i 

1X5.51 

1X5.53 

■■>«5.54 

185.55 

1X5.57 
1X5. 58 
1X5.59 
185.60 
185.61 

1X5.62 
1X5.63 
1X5.64 
185.65 
1X5.66 

185.68 

185.69 

1X5.70 
1X5.71 
1X5.72 

'5*5-"3 
i«5-4 
185.75 
185.76 
185.78 

185.79 

1X5. Xo 

;  185.81 

185.82 

;  i^H 

\   i»5-84 


r. 

12" 

13 

0 

140 

16 

( 

M. 

Dirr.  1". 

M. 

DifT  1'. 

M, 

lit!.  1". 

>i 

Mti  r 

0' 

7.91  1845 

85.84 

8.582146 

1X6.56 

9,255120 

X-.35 

9.93C9X4 

88. Id    1 

1 

7.922995 

185.86 

8.593340 

lXh.5' 

9.26(1360 

'87.34 

9.9422-4 

XX. 18     1 

'Z 

7-9  34 '47 

85.S7 

X.604535 

1X6.5X 

9,277601 

87-35 

9  9^35's 

SX,19     1 

:i 

7.945300 

85XX 

X.O15730 

lXh.59 

9  288812 
9,3000X5 

X-,3- 

9,964X5- 

XS.ii      , 

1 

795*'453 

85.X9 

8.626926 

1X6,61 

87,38 

9.976149 

|XS,21 

.*> 

7.967606 

85.90 

X. 63X123 

1  ^6.62 

9.311  32X 

X-.4C. 

9-987+43 

XX.  ■•1 

41 

7.97X76. 

1X5.91 

X.6A9310 
X.06051X 

i)s6.63 

9-3ii>7S 

'87.41 

9.99X-3S 

XX,2- 

7 

7.9X9916 

8  5. 9  J 

1X6.64 

9.333X17 

X-.42 

10.01003  3 

iXX.iii 

H 

8.00.072 

185.93 

8.671717 

1X6.66 

9.345063 

8'.44 

10.02132c) 

XX   2X 

« 

S. 012228 

185.95 

X. 682917 

186.67 

9.356310 

87.45 

10.032626 

1XX.29 

Hi 

X.023385 

185.96 

8.694,17 

1X6,68 

9.3675^7 

187.46 

10.043924 

IXX.3I 

It 

8.034543 

8597 

X. 705318 

186.69 

9.378805 

,X-.4X 

loo55.'.23 

SS, 32 

I'i 

X. 045702 

1X5. 9X 

8.716510 

186.-1 

9.390054 

87.49 

10,06(1523 

'88.34   ' 

III 

X. 056X61 

1X5.99 

8.727723 

1X6.72 

9.401304 

87.50 

10.077S13 

XX.  3  5      ' 

II 

X. 06X021 

r  86.00 

X.738927 

186.73 

9.412555 

1X7.52 

10.0X9125 

'88.37     ^ 

15 

X. 079181 

1 86.02 

8.750131 

186.74 

9.423X06 

87-53 

10.100427 

,xx.38  ; 

Kl 

8.090343 

86. 03 

X.76I  336 

1X6.76 

9.43505S 

8-. 54 

10.1 11730 

XX. 39     \ 

IT 

S. 101505 

186.04 

8.775541 

1X6.77 

9.446311 

X7,56 

10.123035 

X8.41 

IN 

X.1I266X 

S6.05 

S. 7X3748 

186,78 

9-457^65 

87-57 

10.134140 
10.145646 

'l:■/■^*  i 

li) 

X. 123X31 

186.06 

8-794955 

186.79 

9.46XX20 

1X7.59 

'88.44  ; 

'M 

X.  134995 

18607 

8.X06163 

1X6.X1 

9.4S0076    ' 

iX7..o 

10.1 5695- 

88.45  ; 

'Z\ 

8.146160 

186.09 

8.817372 

186,82 

9.491332 

1X7.61 

10.  l682<'0 

8X,47 

'ii 

8.157326 

86.10 

X.X2X5X2 

l8(>.X3 

9.5025X9 

1X-.63 

10.17956X 

iXX,48     , 

•i:t 

X.  10X492 

rX6.ii 

8.839792 

1X6.84 

9.513847 

1X7.64 

I0.190X7S 

lXX,5o 

'it 

X.179659 

86.12 

8.851003 

186.86 

9.525106 

187.65 

10.202188 

188.51 

'z:. 

8.190X26 

86.13 

X.X62215 

186.X7 

9.536366    ■ 

1X-T.67 

10.213499 

188.53 

w 

8.201995 

1X6.15 

X.X73+27 

1 86.88 

9  ^47626 

1X7.68 

10.224X12 

'^8.54 
XX, 56 

27 

X.2I3I64 

1X6.16 

X.XX46'i 

186,90 

9.5SXXXX 

187.70 

10.236125 

w 

X. 224334 

X6.17 

8.895'  I. 

186.91 

9.570150 

187,71 

10.247439 

88.57 

•i\i 

X. 235504 

186.18 

X. 907070 

186.92 

9.581413 

187.72 

10.25X753 

88.59 

M) 

8.246675 

186.19 

8.918286 

186.93 

9.592676 

1X7.74 

10.270069 

8X.60 

:n 

8.257847 

1X6.20 

S. 929502 

186.95 

9.603941 

'87-75 

10,281386 

XX. 62 

\M 

X  269020 

186.22 

8.940719 

186.96 

9.615207 

87.77 

10.292703 

XX. 63     ' 

:<:< 

8.2X0193 

186.23 

8.951937 

186,97 

9.626473 

187.78 

10.304011 

XX. 65     , 

:m 

X. 291  367 

186.24 

8,963156 

186.99 

9.637740 

187.79 

10.315341 

88.66 

».-> 

X. 30254* 

186.25 

8.974376 

187.00 

9.64900? 

187.81 

10.326661 

88.68     ! 

3a 

S.3137I7 

186.26 

8.985596 

187.01 

9.660277 

87,82 

10.337982 

188,69     1 

:i7 

X.324X93 

186.28 

8.996817 

1X7,02 

9.671547 

87-84 

10.349304 

XX. 71     i 

;w 

X. 3  3(1070 

86.29 

9.00X039 
9.019262 

187.04 

9.682X17  : 

Sf.X5 

10.360627 

88.72    I 

39 

X.347248 

86.30 

187.05 

9.694088 

X7.86 

10.371951 

88.74  : 

10 

8.358426  ' 

186.31 

9.030485 

187.06 

9.705361     : 
9,716634    t 

187.88 

10.383275 

88.75 

11 

8.369605 

86.32 

9.041709 

1X7.08 

87.89 

10.394601    , 

88.77 

\'l 

8.3X0785 

86.34 

9.052934 

9.064160 

187.09 

9.727908    : 

87.9, 

10.405927 

88.78 

13 

8.391966 

186.35 

1S7.10 

9-739'82    1 

87.92 

10.417255 

88, 80 

II 

X.403I47 

86.36 

9.075387 

187.12 

9.750458 

87  93 

10.428583 

88, 81 

l.-i 

8.414329 

86.37 

9.086614 

1X7.13 

9.761734    ; 

187.95 

10.439912 

XX. 83 

Kl 

8.425512 

X6.38 

9.097842 

187.14 

9  773°«»    i 

87-96 

10.451242 

XX, 84 

47 

8.436695 

86.40 

9. 1 0907 1 

1X-.16 

9.784290 

87.98 

10.462573 

XX. X6 

48 

X.447879 

86.41 

9.120301 

187.17 

9795569    j 

87-99 

10.473905 

XX. 87 

41) 

8.459064 

86.42 

9.I3153I 

187.18 

9.806X49    ; 

88.00 

10.485238 

XX. X9 

5(> 

8.470250 

86.43 

9.142763 

187.20 

9.X18129    ' 

88.02 

10,496572 

XX, 90 

51 

8.481436  : 

86.45 

9- "53995 

187.21 

9.X29410    ' 

88.03 

10.507907 

XX. 92 

5'i 

8.492623 

86.46 

9.165228 

187,22 

9.X40693 

88.05 

10.519242 

88-93 

33 

8.503811  i 

8647 

9.176462 

1S7.23 

9.851977    i 

88.06 

10.530579 

^!!-95 

34 

8.515000 

86.48 

9.187696 

187.25 

9.863261 

88  08 

10.541916 

88.97 

55 

8.5261X9  1 

86.49 

9.198931 

187.26 

9.874546 

88.09 

10.553255  ' 

88,98 

30 

8-537379    i   " 

86.51 

9.210167 

187.27 

9.885832 

88.10 

10.564594     1 

89,00 

57 

8.548569  !  1 

86.52 

9.221404 

187.29 

9.897118       1 

88,12 

IO-575934 

89.01 

3N 

8.559761 

86.53 

9.232642 
9.243880 

187.30 

9.908406       1 

88.13 

10.587276       1 

89.03 

59 

8.570953     1 

86.54 

187.31 

9.919694       J 

88.15 

10.598618   1 

89.04 

eo 

8.582146     1 

86.56 

9.255120 

187.33 

9.930984       1 

88.16 

10.609961    1 

89.06 

1 

56» 


TABLE  VI. 

For  fiiuliinf  the  Tnu>  Anomiily  or  tin-  Tinu'  fr  »iii  llie  IVrilu'lioii  in  ii  I'lirabolie  Orhii. 


i 

16 

0 

17" 

18 

19 

M. 

11.667850 

0 

M. 

wir.  1". 

M. 

11.191177 

WIT.  1". 

M. 

1 1.978 1 62 

IHII.  1' 

iwir. '. ' 

lo.fioy<^6i 

189.06 

190.02 

191.04 

.91.13 

1 

lo.hll^o^ 

189.07 

11.303679 

190.03 

II.989(>15 

191.0(1 

12.679379 

19*.  J  5 

u 

i'..f'5it)4'; 

1 89.09 

11.315082 

190.05 

12.001089 

njl.o.'! 

1 2.090908 

»9i  1" 

» 

io.f)4V;95 

189  10 

II  326485 

190.07 

11.012554 

191.09 

11.701439 

IV  2. 19 

'      4 

io.<.55j4i 

189.12 

11. 337889 

190.08 

11.0140x1 

191.11 

11.713970 

I9i»»i 

'      5 

lofifififiyo 

1X9.14 

11.349295 

190  10 

11.035488 

191.13 

11.715503 

192  21 

!f 

189.15 

1  1.360701 

1 90. 1  2 

ll.o4(u^5(i 

191.15 

12.737037 

19124 

7 

189.17 

11.372109 

190.13 

ii.05.>-:4i5 

191. 11' 

i2.74X';73 

192  l(> 

N 

lo. 7007:5  »l 

189.18 

II. 3X3517 

190.15 

12.-j69,-;9(i 

191.18 

12.760109 

192. JS 

» 

io.7iioyo 

189.20 

11.394927 

190.17 

12.0X13(17 

191.20 

12.771646 

192.3, 

lO 

10.71344.1 

189.21 

11.406337 

1 90. 1 8 

1  2.092X40 

191.22 

12.783185 

192.32 

1  1 

lo.7U79i 

1X1J.23 

11.417749 

190.20 

12.104313 

191.24 

12.794714 

192.34 

l-^ 

10  74614'; 

1X9  24 

1  1.429161 

193.22 

12.115788 

191.25 

12.X06265 

l92.-,(. 

i:i 

io.7S7?Q5 

1 89,  id 

II.44C575 

190.23 

12.117264 

I9'^7 

l2..'<l78o7 

192.37 

1 1 

I0.76S86I 

1X9.2X 

11.451989 

190.25 

12.138741 

191.29 

12.819350 

192.39 

l.-> 

10.78011)! 

1 89. 19 

11.463405 

190.27 

12.150219 

.91.31 

11  840X94 

192.41 

10 

10.791  i7h 

189.31 

il.474'<ii 

190.28 

I2.i6i69,s 

191.32 

11.852440 

192.43 

17 

10. Solves 

189.32 

11.486239 

190.30 

12.173178 

191.34 

i2.S(i3986 

'924^ 

IH 

in.!<l4i<>^ 

18934 

11.497657 

19c  32 

12.184659 

191.3(1 

II-X75S34 

'9247 

lU 

io.Xi^655 

1X9.35 

11.509077 

190.33 

12.I96141 

19'  I" 

12.887081 

192.49 

w 

10  >'^7oi7 

1X9.37 

11.520497 

190.35 

12.207624 

191.^0 

12.89X631 

192.51 

'il 

io.K4)i^Xo 

189.39 

11.531919 

190.37 

12.219108 

191.41 

12.910183 

192.53 

'  ut 

ic.S5,)'44 

189.40 

"•';4iuj 

190.39 

12.230594 

>9'43 

12.911736 

'9255 

•i:t 

10.S71  loS 

189.41 

11.554765 

190.4c 

1  2.2420X0 

191.45 

12.9332X9 

192.56 

'Zl 

io.S)!z474 

1S9.43 

1 1.566190 

190.42 

I2.25356X 

191.47 

12.944X43 

192.58 

•zr, 

lo.S9^S40 

189.45 

11.577'iK' 

190.44 

12.265057 

191.49 

12.956399 

191.(10 

'M 

10.90^208 

189.47 

ii.5'<924» 

190.45 

12.17654(1 

191.50 

12.967956 

192  62 

'i7 

10916^76 

1X9.4X 

1 1.600470 

190-47 

12.28S037 

191.52 

12.979514 

192.64 

'iH 

1^.92-946 

1X9.50 

1 1.611899 

190.49 

12.299529 

191.54 

12.991073 

191.66 

'Z» 

loy393i(> 

1 89. 5 1 

ii.623r-!« 

190.50 

12.311011 

191.56 

13.002633 

191.68 

■M 

10.9^06X7 

189.53 

11.634759 

190.51 

11.311516 

191.58 

i>oi4i95 

192. "O 

:ti 

I0.9620VJ 

1X9.55 

11.646191 

190.54 

11.334011 

191.60 

i3-o»S7';7 

192.72 

•,ti 

'0  971413 

1S9.56 

II.657024 

190.5(1 

12.345508 

191.61 

13.037321 

192.74 

:i:i 

10.9S4S07 

1 89. 58 

11.6(19057 

19057 

11.357005 

191.63 

13.048XX6 

192.76 

:ii 

1 0.9961  Si 

1X9.59 

1 1.680491 

190.59 

11.36X503 

191  65 

13.060451 

192.78 

:i5 

1 1.0075^8 

1X9.6 1 

1 1. 691918 

190.61 

11.3X0003 

191.67 

13.071019 

192.80 

:itt 

1 1.01X9;,- 

1X9.63 

I'. 703365 

190.62 

12.391504 

191.69 

13.0835X7 

I92.S2 

:j7 

11.030515 

1X9.64 

11.714803 

190.64 

1  2.403006 

191.70 

13.095157 

I9;.S3 

:iH 

1 1.041691 

1X9.66 

ii.7i'''i4i 

190.66 

12.414509 

191.72 

13.106717 

192.S5 

:w 

11.0^5071 

189.67 

ii.JVMi 

190.68 

12.42601  3 

191.74 

13.118199 

I92.il7 

10 

11.064453 

189.69 

1 1. 749123 

190.69 

i»-4r5'7 

191.76 

13.119871 

192. 89 

11 

11.075X3? 

189-71 

11.760565 

190.71 

12.449023 

191.78 

13.141446 

192.91 

IJ 

11.0X721X 

189.71 

11.772008 

190.73 

1 1.460531 

191X0 

13.1530^2 

192.93 

,  r.i 

11.09X602 

189.74 

11.783452 

190.74 

12.472039 

191.81 

13.164598 

192.95 

II 

1 1.1099X7 

189.76 

11.794897 

190.76 

11.4X354X 

191.83 

13.176176 

192.97 

;     15 

11.121371 

189.77 

11.X06344 

190.78 

11.495059 

191.85 

13.187755 

192.99 

1(( 

11.131759 

1X9.79 

11.X17791 

19  '.Xo 

11.506^71 

191. 87 

i3-'99335 

193.01 

1    '»7 

u.i4+'47 

1X9.80 

1 1. X 292 39 

190.81 

11.518083 

191.X9 

13.210916 

193.03 

i    1?. 

11.155536 

189.X1 

II.X406X9 

190.83 

12.519597 

191  91 

13.212498 

193.05 

1    4U 

11.166925 

189.84 

II. 852139 

190.85 

1  2.541  ■ '  ^ 

19IV3 

13.234082 

193.07 

!    50 

II. 17X316 

189.85 

11.863590 

190.87 

12.552628 

191.94 

13.145667 

193.09 

'    51 

1 1.1X970X 

189.87 

I1.X75043 
II.X86496 

190.88 

11.564145 

191.96 

I3'2>7i53 

193.11 

i    52 

11.101100 

1X9.89 

190.90 

12.575664 

191.98 

J  3.168840 

193.13 

1  5:i 

11.211.194 
11.213X89 

1X9.00 

11.897951 

190.92 

11.587183 

192.00 

15.2X0428 

193.15 

1    54 

189.91 

11.909407 

190.94 

11.598704 

192.02 

1 3.. ••.910 1 7 

193.17 

55 

11.235184 

189.93 

11.910863 

190.95 

12.610225 

191.04 

13.303608 

193.19 

50 

II. 246681 

189.95 

11.931321 

190.97 

11.611748 

192.06 

13.315100 

193.21 

i    57 

•1.258078 

1X9.97 

11.943780 

190.99 

12.633171 

192.07 

13.316793 

193.23 

5H 

11.269A77 

189.98 

II-95S159 

1 9 1. 01 

11.644797 

192.09 

13.338387 

193.25 

50 

11.280876 

190.00 

1 1.966700 

191.01 

12.656313 

1 92. 1 1 

13.349982 

193.27 

00 

11.191177 

190.01 

11.978161 

191.04 

11.667850 

192.13 

13.361579 

193-^9 

570 


ilxilic  OrMi. 


TABLE  VI. 

Kor  liixliiiK  tin-  Tnio  Anoiimlv  or  ilif  Tim*  from  the  IVrihi-lioii  in  a  I'lirnlxilic  OrliiU 


19° 

•      1 

.  1 

Wif.  V 

7«5° 

,.,!.,, 

9179   , 

\ ')*.>;, 

090S   1 

\<tlt- 

i4V; 

IV2.|.> 

%'}7o 

Hjvi.zi 

55°1 

i.>2  j: 

7057 

I.;;  24 

XS71    . 

|.J21(. 

oioi; 

ii;2  2S 

l()4h 

I.J2.V 

^S, 

192.12 

4714 

'9i  u 

(il6^ 

IV2.V- 

7X07 

'yi57 

915° 

192.}.; 

0894 

192.41 

2440 

192.4, 

)V>'<6 

";m? 

'ilU 

192.47 

l7ol<» 

192.4.J 

,X6',2 

192.51 

oiS, 

192.5; 

117,6 

I9i  S? 

5,2X9 

192.5(1 

^"41  : 

19251! 

5 ''199 

192.60 

>79^h 

192  1-1 

/y-iH 

192. ^4 

»i"71 

19266 

u(),3   1 

192.68 

1419?  i 

192.70 

157S7 

192  '2 

i71^«    . 

192-4 

vssxfi  ; 

192.76 

.0452  , 

I92.7S 

72019 

192.^0 

<1'i><7 

.92  S2 

>siS7 

I92.S, 

36727 

I9I..S5 

18299 

fV-*7 

29S72 

192. 89 

M44''    I 

192.91 

S,022 

192.95 

VH'^** 

192.95 

76176 

191.97 

»7755 

192.99 

)9n? 

195.01 

1 09 1 6 

l95-°5 

21498 

195.05 

54C82 

195.07 

j<;667 

193°') 

^-21;? 

195.11 

(1SS40 

195  15 

S0428 

195.15 

[>20I7 

195.17 

0-^608 

195'9 

1^200 

195.21 

26791 

")•,•-) 

1«1«7 

195.25 

49981 

,    19527 

61579 

1   1 93.^9 

0 

20' 

21 

^ 

22 

23 

0 

1,  5''«f79 

IMtT.  1'. 
193.19 

M 

14.059591 

IMIT    1". 

194  51 

M. 
14.762133 

IHir.  1" 

M. 

IMir  1". 
•  97- 1 7 

195.80 

•5469459 

1 

M?71'77 

193   31 

l4.o7iihi 

•94  51 

•  4771''82 

19583 

15.481290 

197  19 

'■i 

15  584776 

•9111 

14.081955 

'94  5  5 

•  4  7''5632 

195.85 

15.493122 

197  21 

:i 

15.396576 

'93-3S 

14.094608 

'94  57 

•4-7971X4 

195.87 

15.5C4956 

'97  24 

•1 

"  3.407977 

«9V17 

14.106283 

•94  59 

14.809137 

195.89 

15  516791 

197.26 

5 

15.419580 

191  19 

14.1  1-960 

194.61 

14.810891 

•95  9^ 

15.52X627 

•  97  28 

0 

n-4i"'<i 

19341 

I  J  129(13- 

194-64 

I4X51(.47 

•95  94 

15  540465 

19'  3> 

7 

1544*788 

'91-<1 

14.141516 

l94-''6 

"4-'<-<44    3 

195  9'' 

l5-5>25t4 

'9-53 

H 

'3.454194 

'9145 

14.152996 

194-68 

14  85(  161 

'95  9X 

15  564144 

'97  35 

M 

i5.4h6ooi 

>91  47 

14.164677 

194.70 

14.86r92l 

19(1.00 

I5-5";9X6 

19-58 

10 

15.477^10 

•91  49 

14.176360 

•94  71 

I4.X79682 

196.03 

15.587850 

197  40 

1 1 

1 5  489220 

195.51 

14.188044 

'94  74 

14.8,^1444 

19(1.05 

I5.59V''75 

'9743 

I'-i 

i5.5'o85i 

•93  53 

14.199719 

19  J  -6 

14  9c 5 208 

196.07 

15  611521 

•9745 

1:1 

•  i  512443 
15.524056 

•93-55 

14  21  1415 

194.78 

'4  9 "4971 

196  09 

15.623569 

'97-47 

1  1 

•91  57 

14.213103 

i94-«^ 

14.926739 

196.12 

15.635218 

197.50 

l<'i 

"3.515''7i 

"91  59 

14.234791 

194.83 

14.918506 

196.14 

15.647068 

•  97-5»     ' 

Ml 

'3-54'287 

19361 

14.2464X1 

194K5 

14.9^0275 

196.16 

1  5.658920 

'97-54 

1 « 

15.558904 

193.63 

■  4258174 

194.87 

14.962045 

196.18 

1  5-670773 

•97  57 

IK 

15.570522 

195.65 

14.269867 

194  89 

14.973X17 

196.20 

15.6X2628 

•  97-59      , 

IW 

15.582141 

193.67 

14.281561 

194.91 

14.985590 

19(1.23 

15.694484 

197.61 

W 

l3-5917<i2 

193.69 

14.293156 

•9491 

•4-9973'' 5 

196.25 

15.706342 

1976A 

197.66 

'il 

15.1.05585 
15.617006 

193.71 

•4-304953 

'9495 

1  5.009140 

19(1.27 

15.718201 

'i'i 

•91-71 

14.316(151 

194.98 

1  5  020917 

196.30 

15  71o:'''l 

197.69 

•ill 

15.628631 

•91-75 

14.328350 

195.00 

1 5.o326<i6 

196.31 

15.741923 

15.753786 

197.71 

•it 

1  5.640256 

•91-77 

14.340050 

195.01 

15.044475 

19(1.34 

•97-7  3 

•-!.■> 

156518S3 

•9179 

14.551752 

195.04 

15.056256 

196.36 

15.765651 

197.76 

•m 

15.663511 

193.81 

•4-163455 

195  06 

1 5.068039 

196.39 

'  5-7775 '  7 

197.78 

•i' 

15.67^140 

193.83 

I4-I7';i^9 

195.08 

15.079823 

196.41 

15.7X9I-X5 

197.80     1 

'iX 

15.686770 

193.85 

14.386865 

195.10 

1  5.091608 

196.43 

15.801254 

'97X3 

'M 

1  5.698401 

193.87 

14.398572 

195.13 

15.103394 

196.45 

15.813124 

197.85 

m 

13.710034 

193.89 

14.410280 

195.15 

15.115182 

196.48 

15-824996 

197.88 

:<i 

15.721668 

193.91 

14.421990 

•95  '7 

15.126971 

196.50 

15.836870 

197-90 

;ji 

'3  ■'31101 

•93  91 

14.433700 

195.19 

I5.i5'<76i 

196.52 

15.848744 

197.92 

WW 

'1'44940 

•91-95 

14.445412 

195.11 

i5.i>o554 
15.161348 

196.54 

15.860620 

'97-95 

:<i 

•3-75(>577 

•91-97 

14.457126 

195.23 

196.57 

15872498 

•97-97 

:t.-> 

15768J16 

193.99 

14.468841 

195.26 

15.1-4142 

196.59 

15.884377 

198.00 

•M\ 

15779856 

194.01 

14.480557 

195.28 

15.1X591X 

196.61 

15.S96258 

198.02 

;»7 

I5.79i49« 

194.03 

14.492274 

195.30 

15.19771'' 

196.64 

15908 140 

198.04 

:w 

13.803140 
13.814784 

•94-05 

14.503991 

195.32 

15-209535 

196.66 

15.920025 
15.931908 

198.07 

\Vi 

194.07 

14.515711 

•95-14 

15.111335 

196.68 

198.09     i 

to 

1 3.S26429 

194.09 

'4-527414 
•  4-559»56 

•95-16 

•5-233«37 

196.70 

•5  941794 

19X.11 

II 

15.X38075 

194.11 

•95-19 

1  5-244940 

196.73 

15.955681 

19X.14      [ 

li 

15.849713 

194.14 

14.550880 

'954' 

'5-2^6-44 

196.75 

15.9675-1 

19X.17 

III 

15.861371 

194.16 

14.561605 

•95-43 

15.26X550 

196.77 

15.979462 

19X.19 

II 

15.873011 

194.18 

•4-57411' 

•95-45 

15.280357 

196.80 

'5-99' 154 

198.21 

Vt 

15.88.673 
13896315 

194.20 

14.586059 

•95-47 

15.191165 

196.82 

16.603148 

19X.24  : 

la 

194.22 

i4-5977'*>' 

195.50 

15-301975 

196.X4 

16.015143 

198.16 

IT 

1 5-907979 

194.24 

14.609519 

195.52 

15.315786 

196.87 

16.027039 

19X.29 

IK 

15.919634 

194.26 

14.621250 

•95-54 

15.32-599 

196.89 

16.03X937 

19S.51 

li» 

13.931290 

194.28 

14.631983 

195-56 

'  5-1194' 1 

196.91 

16.050X36 

19X.34 

:>() 

15.942948 

194.30 

14.644718 

195.58 

15.351228 

196.94 

16.062737 

19X.36  1 

51 

13.954606 

194.32 

14.6^6453 

195.60 

15.363045 

196.96 

16  074639 

198.38 

5i 

1 5.966166 

•94-14 

14.668190 

195.63 

'5-174X''1 

196.98 

16.0X6543 

19X-4' 

5:« 

>  19779*7 

194.36 

14.679919 

195.65 

15.386683 

197.00 

16.098449 

198.43 
198.46  i 

51 

13.989590 

194.38 

14.691668 

195.67 

15.398504 

197.03 

16.110355 

5.-> 

14.001154 

19441 

14.703409 

195.69 

15.410326 

197.05 

16.122263 

198.48 

50 

14.012919 

•9441 

14.71515^ 

195.71 

15.422150 

19T.07 

16.134173 

i9«-5' 

57 

14.024585 

•94-45 

14.716895 

•95-74 

'5-411975 

197.1c 

16.1460X4 

198.53 
•9X-56     1 

5H 

14.036151 

•94  47 

14.73X640 

195.76 

15.445X02 

197.12 

16.157997 

51) 

14.047911 

•94-49 

14.750386 

195.78 

15457630 

197-14 

16.169911 

198.58 

GM 

14.059591 

194.51 

14.761133 

195.80 

'5-4<'9459 

197.17 

16.181826 

19X.60 

571 


TABLE  VI. 

Vor  (\ni\\n(t  llu-  Triii>  Anoiii.ilv  nr  llir  'rime  rniiii  the  I'lrilitliiiii  in  ii  I'liriilHilic  Orliil. 


t'. 

24 

M. 

') 

25 

M 

<J 

lilff.  1". 

26 

27 

\j 

lh(l    l". 

M 

IHIT.  1". 

M. 



Wff.  1", 

o 

i(<.lXl8l6 

I9X.60 

16.X99499 

100. 1 1 

17  621747 

201.70 

I«35ill47 

»C337 

1 

i(..lyr41 

I9X.63 

16.911507 

200.14 

17.63^850 

201.73 
201.70 

18.3(14050 

103.40 

'i 

Id.l  j?<)hi 

19><.'<5 

16,913^16 

200. 1 7 

17.646954 

18.3-6255 

20341 

!      3 

16.11:5X1 

I9X.6X 

16.935517 

200.19 

17.6^9060 

201.78 

18.3XX461 

*f^3  45 

1 

Ifi. 21950^ 

I9X.70 

16.947539 

100.22 

17.671168 

201.81 

18.400669 

1034X 

» 

ifi.i4i4i() 

19X71 

l'^";595  5  3 

100.24 

17.6X317X 

201.84 

iX. 411879 

103.51 

1     n 

1''  i>n?° 

t')^-7> 

16.9715(18 

200.27 

176953X9 

201.87 

18.425090 

103  54 

7 

1(1.1(1^176 

I9X.7X 

16.9X35X5 

200.30 

17.707501 

201. 89 

iS. 437303 
iX. 449518 

103  57 

H 

l<i.i7''»o4 

19S.X0 

16.995(04 

200.31 

17.719616 

101.91 

203  ;,, 

« 

1(1.1X91  ^^ 

I9X.S3 

17.007614 

100.35 

'7-73'73» 

101.95 

i'l-4f"735 

203  62 

10 

Id.  V)  106^ 

19S.S5 

17.019646 

200.37 

'7-743'<50 

201.97 

i-"<-473953 

203,65 

1  !.'. 

i(..';ii99<; 

19X.XX 

17.0316(19 

100.40 

17.751:969 

102.00 

18.4X6173 

103. 6X 

!    vz 

l('^i49iX 

19X.90 

I7.0436ri4 

100.43 

17.76X090 

202.03 

1849X395 

103.71 

i    i:t 

1(1.  n''^''^ 

19.^.93 

i7.0557:.o 

200.45 

i7.78o-.<.i3 

202.06 

1X.51061X 

203.74 

1    It 

><'U«797 

19X.95 

17.067748 

2C0.48 

'7.79»337 

202.08 

18.521843 

203.77 

ir> 

16.360737 

198.97 

17-079777 

200.50 

17.804461 

202.11 

18.535070 

203.80 

10 

16  371676 

I99.OU 

17.091  Xc'X 

200.53 

17.X16590 

202.14 

18.547199 

203  82 

17 

16.3S4617 

199.01 

17.103X41 

200.56 

17.!<1X7'9 

202.17 

1X5 595 29 

203X5 

IH 

16.3961559 

« 99-05 

17.115X75 

200. 58 

17.X40X50 

202.19 

18.571761 

203. 88 

lU 

16.40X503 

199.07 

17.127911 

200.61 

17.X52982 

202.22 

18.583995 

203.91 

W 

16,41044s 

199.10 

17.1399.^8 

200.64 

17.865116 

202.25 

18.596230 

203.94 

Ul 

16.431395 

199.11 

I7.I5'9'<7 

200.66 

17.877151 

102.28 

1 8,60X467 

IC3.9-- 

Ti 

16.444343 

199.1.- 

1 7.1 6401*' 

200.69 

17.8X9389 

101.30 

iX. 620706 

IC4.00 

T.t 

16.456192 

199.17 

17. 176070 

200.71 

17.901528 

202.33 
201.36 

iX. 632947 

104.03 

'H 

i6.4()Xi43 

199  10 

17.188114 

200.74 

17.913669 

iX. 645190 

104.05 

ar. 

16.4X0196 

199.21 

17.1001,59 

200.77 

17.925811 

101.39 

iX. 657434 

204.08 

•M 

16.4911  51 

199.15 

I7.2i2106 

200.79 

'7-';3-'955 

101.41 

iX. 669679 

204.11 

'i7 

16. 50  J,  107 
16,1;  16064 

199.17 

17.224154 

200.X1 

17.950101 

201.44 

18.681927 

104.14 

UH 

199.30 

I7.13630A 

I7.24«356 

200. 85 

17.961248 

202.47 

18.694177 

104.17 

'^U 

16.51X021 

199.33 

200.87 

'7-974397 

202.50 

IX.70641X 

104.20 

:m» 

16.539983 

«9«MS 

17.260409 

200.90 

I7.9«6548 

202.52 

18.718680 

104.13 

:ti 

16.55194c 
16.5(13908 

199. 38 

17.2-1464 

100.93 

17.99X700 

201.55 

18.730935 

104. :(, 

:t'i 

199.40 

17.2X4520 

200.95 
200.98 

18.010854 

101. 5X 

IX.743I9I 

204.19 

:i:i 

16.575X73 

»V'r43 

17.396578 

18.013010 

202.61 

'8-7  5  5449 

204.31 

31 

16.587X39 

199.45 

17.308637 

201.00 

18.035167 

202.64 

18.767709 

204.35 

'    35 

16.599807 

199.48 

IT. 310698 

101.03 

18.047316 

201.66 

18.779971 

204,37 

;  30 

16.611776 

199.50 

17.332761 

201.06 

1X.0591X7 
iX. 071649 

102.69 

18.791234 

104.40 

37 

16.613747 

'99-53 

17.344X15 

201.08 

202.72 

18.804499 

204.43 

1     3N 

16.635719 

>99-55 
199.58 

17.356X91 

201. II 

1X.083X13 

202.75 

18.816767 

204.46 

'    30 

1 

16.647093 

17.368959 

201.14 

18.095979 

202.78 

18.829036 

204.49 

40 

16.659669 

199.60 

17.381018 

201.16 

1X.108146 

202.80 

18.841305 

204.51 

!     41 

16.671646 

199.63 

17.393098 

201.19 

iX. 120315 

202.83 

18-853577 

204.55 

!    4'i 

16.683614 

199.65 

17.405171 

201.12 

1X.I32486 

202.86 

18.X65X51 

204.58 

43 

16.695604 

199.68 

17.417245 

201.24 

18.144658 

202.89 

18.X7X127 

204.61 

44 

16.707586 

199.70 

17.419310 

201.27 

18.156831 

202.92 

1 8.x  90404 

204.64 

45 

16.719569 

>99-7? 
199.76 

>  7-44 '397 

201.30 

18.169008 

202.94 

18.902684 

204.67 

40 

16.731553 

•  7-4.5  347f' 

201.31 

18.181186 

202.97 

18.914965 

204.70 

47 

16.743539 

199.78 

17.465556 

201.35 

iS. 193  365 

203.00 

18.927247 

204.73 
204.70 

48 

16.755527 

199X1 

17.477638 

201.38 

iS. 105546 

103.03 
103.06 

18.939531 

40 

16.767516 

199.83 

17.4X9721 

201.41 

18.117728 

18.951818 

204.79 

50 

16.779507 

199.86 

17.501807 

201.43 
201.46 

18.119911 

203.08 

18.964106 

204.81 

51 

16.791499 

199.88 

17. 5«  3894 

18.242098 

203.11 

18.976396 

204.84 

52 

16.X03493 
16.815488 

199.91 

17.5159S2 

201.49 

iX. 154286 

203   14 

18.9886X7 

204.87    , 

S3 

>99  94 

17.53X072 

101.51 

18.166475 

203.17 

19.000981 

104.1,1    1 

54 

16.827485 

199.96 

17.550163 

101.54 

18.178666 

203.20 

19.013276 

204.93    , 

55 

.6.839484 

199.99 

17.562257 

201.57 

18.190859 

203.23 

19.025573 

204.96 

50 

16.8514X4 

200.01 

'7-574351 

201.59 

iX. 303053 

203.25 
203.28 

19.037871 

204.99 

57 

16.863485 
16.X75488 

200.04 

17.586448 

201.62 

18.315249 

19.050172 

105.02 

58 

200.06 

17.598546 

201.65 

iX. 327447 

203.31 

19.062474 

205. 0^ 

50 

16.887493 

200.09 

17.610646 

101.68 

IX.339646 

203.34 

19.074778 

105.08 

60 

16.899499 

200.12 

17.622747 

201.70 

18.351847 

203.37 

19.087084 

205.11     , 

672 


liolicOrliil. 


TABLE  VI. 

For  flMilinff  tlu'  Triii'  Anoinnlv  or  tlie  Tiiiio  Irom  ilu-  IVrihi-lion  in  n  I'nrntK.lii'  Orl)il. 


27 


iHiT.  r 


l8+7    1 

»cvr 

P>0     ' 

103.4c 

»li!^ 

10341 

1*4(11 

103  41; 

06  ('9 

1034X 

i«7'J 

103.;! 

1™;° 

103  <+ 

7 101 

103  r 

103  ;., 

•71? 

103  (is 

V^>1 

ioyhf, 

hl71 

103. 6X 

*vr^  , 

103.71 

oMX    I 

203.74 

1S43 

103.77 

^070 

103. So 

7Z'J9 

103  *i 

ySi'J    1 

103. «^ 

I7''i 

203. XX 

vm 

203.91 

(m-\o 

103.94 

«4"7 

103.9-' 

070(1 

204.00 

1947 

104.03 

SI  90 

204.0? 

7414 

I04.0X 

9679 

104.11 

1917 

104.14 

4<77 

104.17 

)64S« 

104.20 

S680 

104.23 

o'>1S 

204:'< 

I'V 

204.29 

5449 

104.31 

.7709 

104.35 

997" 

104.37 

iJiH 

104.40 

>4499 

104.41 

67(^7 

204.4(1 

19036 

■■  204.49 

hMOS 

'  204.51 

;3<!77 

204.55 

.SSsi 

204. 5X 

■XI27 

,  204,61 

(0404 

i  204.64 

n684 

i  204.67 

14965 

204.70 

'7247 

104-71 

19^2 

j    104.7'' 

;iSi8 

,   204.79 

14106 

204.XI 

r6v)6 

204X4 

;S6X7 

:   204.87 

309SI 

204.1, 1 

13276 

i   104-91 

i';?7i 

104.96 

17!*7i 

104.99 

^0171 

205.02 

'2474 

205.05 

7477« 

105. oX 

R7084 

1   105.11 

1: 
0 

28 

') 

29 

30 

31 

J 

M 

19  oH7o»4 

ihir.  1". 
205.11 

M 

19.X18747 

Piff.  1". 

I'a  M. 

iMtr.  1". 

l"«  M. 
1.319  0430 

mrr  v     ' 

106.94 

1.313   3»'49 

44.08 

I 

• 

19.099391 

105.14 

19841164 

106.97 

.313  (1493 

44.06 

.319   3001 
.319   5578 

4291 

'i 

19.1 1 1*01 

105.17 

19X5,583 

107.00 

.313  913(1 

44-'^'4 

41. 89 

:i 

19  114012 

105.10 

i9.X(i()004 

107.03 

.1i4    •">< 

44.01 

.319  8151 

42.87 

1 

19.1 3'' 3>  5 

105.13 

19.878417 

107.0(1 

.314  44^9 

44,00 

.330  0723 

41.8J 

.'» 

19  ivX(i3(, 

105  16 

19  X90X51 

107.09 

1.314   -  ,5^ 

43.98 

1.330  3193 

41.83 

tl 

19. 1(10956 

205  19 

19  9^1279 

207.13 

.314   9(..,(, 

43.96 

.330   5XM 

41. Xl 

I 

19173*74 

105  31 

19  9' 5707 

207.1(1 

■1^5  Mr. 

4194 

.330  8431 

42.80 

H 

19.1X5574 

205  35 

19918137 

107.19 

.315   49">9 

4192 

.331    099X 

41.-8 

it 

19.197916 

105. 3X 

19.940569 

107.21 

.315    7<'04 

43.90 

•31^    15''4 

41.76 

l«» 

19.210240 

105  41 

19953003 

107.15 

1.316   0137 

43.X8 

1-3  r   6119 

42-74 

II 

19.121566 

205.44 

•9  9''5419 

107.18 

.316    lS(i(^ 

43,X(, 

.331    8(193 

41.71 

1'^ 

19.234X93 

205.47 

•  9  9"'7877 

107.31 

.31(1    5  5'>o 

43. X4 

•312    1155 

41.70 

i:i 

19.147111 

105.50 

19.99-J317 

107.34 

.31''    X130 

4i.>;i 

-112     381- 

.3  3:'   '""8 

42.69 

II 

'9*59551 

105.53 

10  001759 

107.3X 

.317   0759 

43.X0 

41.67 

l.'i 

19.271885 

105.56 

10.015101 

207.41 

I.317    3386 
.317   6013 

43.7« 

1.332  8»j 

41.65 

Itl 

M). 1X1110 

105.59 

10.017647 

107.44 

43.76 

.333   149'. 

41.63 

1 1 

19.29(1556 

105.61 

10.040095 

107.47 

.31-'   863X 

41-4 

•Ill    10.,  3 

41.61 

IS 

I9.30SX94 

105.65 

10  051544 

107.50 

.31X    1161 

4171 

"-,   (16011 

42.59 

111 

i9.3n»U 

105.68 

2o.-'i4995 

207.53 

.318    3X8, 

41-70 

1  ,3  9>"4 

4i.5'< 

W 

"9U157fi 

205.71 

10.077448 

107.57 

I.31X   6506 

41.68 

•  314   "71  ' 

41.56 

'ii 

19  1459»o 

ir  ■,- 

10.0X9903 

107.60 

.31X   9127 

41''7 

•114  4271 

42  54 

'ii 

19.35X165 

105. /7 

10.102160 
20.114818 

207.63 

.319    '74*' 

4; ''5 

•314   "Xil 

42.52 

•i.l 

19.370612 

105.80 

107.66 

.319   4364 

43.(13 

•134  934 

41.50 

'it 

19.18;.    . 

105.83 

10.1  27179 

107.69 

.319   6981 

43-'' I 

•13  5    '724 

41.49 

v> 

19.395311 
19.407665 

105. X6 

2o.i3974« 

107.71 

1.319   9597 

41-59 

"•115  4471 

42-47 

w 

105. X9 

10.151106 

107.76 

.320    2211 

41-57 

.335   7010 

42-45 

'i7 

19.410019 

105.91 

10.164671 

207.79 

.310    4815 

41-55 

■il'i  95"7 

42.43 

'iH 

•9412175 

105.95 

10.177140 

107.81 

.320    7438 

41-51 

.33(1  2112 

41.41 

w 

19  444714 

105. 9S 

20.1X9610 

107.85 

.321     0049 

435" 

.336  4656 

42.40 

:m 

•9457094 

106.01 

10.101081 

107.88 

1.321     2659 

41-49 

1.336  7199 

41.38 

:tl 

•9-4('945  5 

106.04 

10.114556 

107.91 

.311     526X 

41-47 

.336  9741 

41.36 

:r^ 

19.4X1819 

206. oX 

10.217031 

107.95 

.3:1     7X75 

41-45 

•117   2283 

42.34 

:i:i 

19.494184 

206.l  1 

10.239510 

107.9X 

.321    0481 

41+1 

•117  41*23 

42.31 

:u 

19.506551 

106.14 

20.2519X9 

108.01 

.322     30X7 

41-4^ 

•337   73''2 

41.31 

X, 

•  9-5i*'9H 

106.17 

20,261471 
20.IT6954 

108.04 

1.322     5691 

43.40 

1.337  9900 

41.19 

:m 

19.531292 

106.10 

108.07 

.322    8195 

43.38 

.338  2437 

42.27 

;n 

19.543664 

106.23 
106.2(1 

20.2X9440 

20X.1 1 

.313    0897 

43.36 

.338  4971 

42.25 

;w 

19.556039 

20.301917 

208.14 

.313     3498 
.323    6097 

41-14 

.338  7507 

42-24 

:iu 

19.568415 

206.29 

10.314416 

108.17 

41-12 

•3  39  004 « 

41..'.2 

M 

19.580794 

106.31 

20.326907 

208.10 

1.323    8696 

41-10 

"•339  2573 

42.20 

II 

•  9. 591 1 74 

106.35 

20.339400 

208.14 

.314     1294 

43.28 

•139  5105 

42.18 

n 

19.605556 

106.38 

20.351895 

108.17 

.324    3X90 

43.26 

•3  39  7'i35 
.340  0165 

42.17 

i:i 

•  9.617939 

206.41 

20.364192 
20.376891 

108.30 

.324    ''4K5 

4324 

42.15       , 

II 

19.630325 

106.44 

108.33 

.324    9079 

43.21 

.340  1693 

42.13 

l.-i 

19.642713 

206.47 

20.389192 
20.401X95 

108.36 

1.325     1672 

4j.11 

1.340  5111 

41.11 

Id 

19.655101 

106.50 

108.39 

.325    4263 

43.19 

•340  7747 

41.10 

IT 

19.667193 
I9.6798.<6 

106.53 

20.414399 

20S.43 

.315     f'854 

41.^7 

.341    0171 

42.08 

IN 

106.57 

20.426906 

108.46 

•125   9441 

41'5 

.341    1796 

42.06 

I'J 

19.691281 

106.60 

20.439415 

108.49 

.316   2032 

43-' 3 

•34^    5  3"9 

42.04 

50 

19.704678 

106.63 
106.66 

10.451925 

108.51 

1.326  4619 

43.11 

1. 341   784' 

42.03 

51 

19.717076 

20.464437 

108.56 

.326  7205 

43.09 

.342  0362 
.342  2882 

42.01 

yi 

«9-729477 

106.69 

20.476952 

208.59 
108.61 

.326  9790 

43-07 

41.99 

53 

19.741879 

106.71 

20.4X9468 

•327   2374 

43.05 

•342   540" 

4«97 

51 

19.754183 

106.75 

10.501986 

108.65 

•327  4957 

43.04 

-3V2  79  "9 

41.96 

55 

19.766689 

106.78 

10.514506 

108.69 

1.327  7538 

43.02 

1.34,  0436 

4»94 

5)1 

19.779097 

106.81 

10.517019 

108.71 

.328  0119 

43.00 

•343  2951 

41.91 

57 

19.791507 

106.84 

20.519553 

108.75 
108.78 

.318   1698 

41.98 

•343  5467 

41.90 

SH 

19.803919 
19.816331 

106.88 

10.552079 

.318   5176 

42.96 

•343  79«o 

41.89 

5U 

106.91 

10.564607 

108.81 

.318  7853 

42.94 

•344  0491 

41.87 

QO 

19.818747 

106.94 

20-S77137 

108.8s 

1.319  0430 

41.91 

1.344  300$ 

41.85 

573 


TABLE  VI. 

For  fimliiiK  the  True  Annmiily  or  thf  'I'liiu-  from  tlie  Perihelion  in  .1  Parabolic  Orliil. 


1 
t 

1       0' 

32 

0 



33 

0 

34 

0 

35 

0 

1.344  3oo<; 

Dlff.  1". 

loK  M. 
1.359    1859 

I)iff.  1". 

log  SI. 

Dlff.  1". 

IngM. 

Diir.  1". 
39.06 

41.X5 

40.86 

'•373 

7251 

39-93 

..387 

941S 

1 

■U4   SSI? 

4i.l'4 

•359 

43'° 
6760 

40.84 

•373 

9646 

39.91 

.388 

1761 

39-'-j5 

'i 

■in  ><'^i5 

41.82 

•359 

40.82 

•374 

2041 

39.90 

.388 

4104 

39-^4 

:i 

■:?n  osu 

41.80 

•359 

9209 

40.81 

•374 

4434 

39.88 

.388 

6446 

3902 

4 

•345   304" 

41.78 

.360 

1657 

40.79 

•374 

6827 

39.87 

.388 

8787 

39.01 

6 

"•345   lU* 

4'-77 

1.360 

4104 

40.78 

'•374 

9218 

39.85 

1.389 

1127 

38.99 

a 

•345   X053 

4«^75 

.360 

6550 

40.76 

•375 

1609 

39.84 

•389 

3466 

38.98 

i     I 

.34(1  oi,-K 

4'-73 

.360 

S995 

40.74 

•375 

3999 
6388 

39.82 

-389 

5804 

38-97 

!     ** 

.346   3061 

41.72 

.361 

'439 

40.73 

•375 

39.81 

-389 

8142 

38.95 

i     1) 

.346    S564 

41.70 

.361 

3883 

40.71 

•375 

8776 

39^79 

•390 

0479 

38-94 

10 

1.346   806^ 

41.68 

1. 361 

6325 

40.70 

1. 3^6 

1164. 

39.78 

1.390 

2815 

3893 

II 

•347  oS<'S 

41.66 

.361 

876(1 

40.68 

.376 

3550 

39^77 

•390 

5150 

38.9, 

Vi 

•U?    io('i 

41.65 

.362 

1207 

40.66 

•376 

5935 

39^75 

-390 

7484 

38. 90 

>    1:1 

•347   55^'3 

41.63 

.362 

3646 

40.65 

.376 

8320 

39^74 

.390 

9817 

3S.88 

i    11 

.347    SoDO 

41.61 

.362 

6084 

40.63 

•377 

0703 

39.72 

.391 

2150 

38.87 

15 

1.34S  05^-7 

41.60 

1.362 

8522 

40.62 

'•377 

3086 

39-7' 

'•39' 

4482 

38.S6 

1({ 

.348   30^2 

41.58 

.363 

0959 

40.60 

•377 

5468 

39.69 

•39' 

6813 

38.84 

17 

.34X    5546 

41.56 

.363 

3394 

40.59 

•377 

7849 

39.68 

•39' 

9'43 

38-83 

IH 

.348   8040 

4'^55 

.363 

5X29 

40';7 

37« 

0230 

39.66 

•392 

1472 

38.82 

lU 

•349  o53i 

4«-53 

.363 

X263 

40.56 

.378 

2609 

39.65 

.392 

3801 

38. So 

ao 

'•349   3013 

41.51 

1.364 

0696 

40.54 

1.378  4987 

39.64 

1.392 

6128 

38.79 

'^i 

•349   S5«3 

41.50 

.364 

3128 

40.52 

.378 

7365 

39.62 

.392 

8455 

38-77 

'Zt 

•349  *<o^^3 

41.4X 

.364 

5559 

40.51 

.378 

9742 

39.61 

-393 

0781 

38.76 

U',l 

.350  0491 

41.46 

.364 

7989 

40.49 

•379 

»li7 

39-59 

-393 

3107 

38.75 

'21 

•35°  1978 

41.45 

.365 

0418 

40.48 

•379 

4492 

39^58 

-393 

543' 

38.73 

25 

'•350  5464 

41.43 

1.365 

2846 

40.46 

'•379 

6866 

39.56 

'•393 

7755 

•J8.72 

'M\ 

•35°  7950 

41.41 

•365 

5273 

40.45 

•379 

9240 

39^5  5 

•394 

0078 

38.7I 

27 

.351   0434 

41.40 

•365 

7699 

40.43 

.380 

1612 

39^53 

•394 

2400 

38.69 

28 

•35«    29'7 

41.38 

.366 

01  25 

40.41 

.380 

3983 

3952 

•394 

4721 

3X.6X 

2U 

•35 «    5399 

41.36 

.366 

2549 

40.40 

.380 

6354 

39.50 

•394 

7041 

3S.67 

■  •  a 

i.3<;i   78X0 

4135 

1.366 

4973 

40.   8 

1.380 

8724 

39^49 

'-394 

9361 

38.65 

1    :<! 

•j52  03''' 

4«^33 

.366 

7395 

40-3 

.381 

1093 

39-47 

-395 

1680 

38.64 

■.i'i 

.3S1   2840 

41.31 

.366 

9817 

4°^35 

.381 

H^l 

39-46 

•395 

3998 

38.63 

1   :!•! 

•352  53'^ 

41.30 

.367 

2238 

40-34 

.381 

5828 

39-45 

•395 

6315 

38.61 

!    :>> 

•351   7795 

41.28 

.367  465/ 

40.32 

.381 

8 '94 

39-43 

-395 

8631 

3S.60 

:i5 

'•353  0*7* 

41.26 

1.367 

7076 

40.31 

1.382 

0559 

39.42 

1.396 

0947 

38^59     i 

:i(i 

•353   ^747 

41.25 

•3'''7 

9494 

40.29 

.382 

2924 

39.40 

.396 

3262 

38^57     , 

i    :i7 

•353    5"! 

41.23 

.368 

19.1 

4028 

.382 

5288 

39-39 

•396 

5576 

38.56 

;w 

•353   7C'94 

41.21 

.368 

4327 

40.26 

.382 

7651 

39^37 

.396 

7889 

38-53 

:iu 

•354  01  "7 

41.20 

.368  6742 

40.25 

.383 

0013 

39-36 

•397 

0201 

38-53 

10 

1.354  ;.6  58 

41.18 

1.368 

9'57 

40.23 

1.383 

2374 

39-35 

'•397 

25  r  3 

38.52 

4! 

■354   5108 

41.16 

.369 

I  570 

40.21 

.3S3 

4734 

39-33 

•397 

4823 

3851 

42 

•354  757** 

41.15 

.369 

3983 

40.20 

•3i^3 

7093 

39^3  2 

•397 

7'3  , 

38-49 

4:t 

•355   004'' 

41.13 

.369 

6394 

40.18 

•3i*3 

9452 

39^3° 

•397 

9442 

38-48 

44 

•355  *5'3 

41. II 

•3  ('9 

8805 

40.17 

.384 

1809 

39^29 

-398 

1751 

38.47 

45 

1.355  49*'° 

41.10 

1.370 

1214 

40,15 

1.384 

A166 
6522 

39.27 

1.398 

>'.o58 

38-45 

40 

•3  55   7445 

41.08 

.370 

3623 

40.14 

.384 

39.26 

•398 

(',-,65 

38-44 

47 

.355  9909 

41.07 

.370 

6031 

40.12 

.384  8878 

39-25 

.398 

86',  ( 

38.43 

4H 

•35''   2373 

41.05 

.370 

8438 

40.  n 

.385 

1232 

39-23 

•399 

<97'' 

3   .4. 

lU 

.356  4836 

41.03 

•371 

0844 

40.09 

.385 

3585 

39.22 

•399 

32' I 

3S.40 

50 

1.356  7297 

41.01 

1.371 

3249 

40.08 

1.38; 

5938 
8290 

39.20 

1.399 

55^ 

38.39 

51 

.356  9758 

41.00 

•37» 

5654 

40.06 

.385 

39.19 

•399 

7887 

5^^I 

52 

.357   i2«7 

40.98 

■37' 

8057 

40.05 

.3S6 

0641 

39.18 

.400 

0189 

38.36 

5:1 

•357  4''76 

40.97 

•372 

0459 

40.03 

.386 

2991 

39.16 

.400 

2491 

38.,- 

54 

•357  7«34 

40.9s 

.372 

2X61 

40.02 

.3S6 

5340 

39-15 

.400 

4791 

38^33 

55 

1.357  9590 

40.94 

'•37?. 

5261 

40.00 

1.386 

7689 

39-n 

1.400 

7091 

38-32 

5A 

.358   2046 

40.92 

•372 

7661 

39^99 

.3S7 

0036 

39^  1 2 

.400 

9390 

38-3' 

57 

.358   A501 
•35>^  6954 

40.90 

•373 

0060 

39^97 

.387 

2383 

39.11 

.401 

1688 

^S'^S 

5H 

40.89 

•373 

2458 
4855 

3996 

•3«7 

4729 

39.09 

.401 

r>^5 

18.18 

5» 

.358  9407 

40.87 

•373 

3994 

.387 

7074 

39.08 

.401 

6282 

38.27 

(M) 

1.359  '859 

40.86 

•■373 

7251 

3993 

'•387 

9418 

39.06 

1.401 

8578 

38.26 

m 


iibolic  Orldt. 


TABLE  VI. 

For  fmdinp  tlie  True  Anoiniily  or  tlic  Time  from  the  Perihelion  in  a  Piiraholic  Orhh. 


35= 


?M. 


Dinr.  1 ". 


9418  1 

39.06 

1761 

39'-^5 

4104 

39-^4 

6446 

3902 

S787  i 

39.01 

1127 

3X99 

3466 

3X.9X 

5«04  ; 

3^-97 

XI42 

38.95 

0479 

3X.94 

281? 

3«93 

5IS0 

3X.9, 

74«4  1 

3X.90 

9817  1 

3X.X8 

2150 

3X.X7 

4482  i 

3X.X6 

6813 

3X.X4 

9143 

3X.X3 

,  1472 

3X.X2 

I  3801 

3X.S0 

.  6128 

38.79 

-  8455 

3>''77 

!  07X1 

3X.76 

J  3107 

3>*-75 

5  543» 

3>'-73 

5  7755 

■jX.72 

^  007X 

38.71 

1  2400 

3S.69 

^  4721 

3X.6X 

h  704' 

3S.67 

^  9361 

3X.65 

;  16X0 

3X.64 

;  399i< 

3X.f,3 

;  f'3'5 

3S.61 

;  «('3' 

3X.60 

'  0947 

3X.59 

)  3262 

3!<-57 

S  S576 

3X.56 

S  7XX9 

1  ^!!-'' 

7  0201 

3i<-53 

7   25'3 

'  3^'5^ 

7  4'<^3 

3X5 1 

7  713-. 

:  ^'^->'! 

7  944^ 

•;X.4X 

if  1751 

i  3^'47 

X  .-.OS 8 

3!<-45 

8  rtift^ 

'  3^-44 

«  X(.',  r 

!  ^U^ 

9  ^97' 

!  3'^-4' 

9  3i'i 

3X.40 

9  5584 

3X.39 

9  7««7 

38.37 

0  0189 

!  3^-3'^ 

0  2491 

i  3X.,- 

0  4791 

1  3>'-3i 

0  7091 

1  38.  V- 

0  9390 

1  3«'3> 

I  1688 

1  38.30 

I  3985 

i  ,8.2X 

I  6282 

i  3*''-7 

I  8578 

1  38. 26 

r. 


'Z 
A 

4 

5 
» 

7 

H 
9 

10 
II 
12 

i:i 
II 

15 

l(i 
17 

IH 
lU 

20 
21 
'i'i 
'iA 
21 

2.-1 

2« 
27 
2H 
2S» 

:<() 
:u 

\n 

:ii 
;{.■> 

:i7 

:iH 
:t!) 

10 
II 
12 
r.i 
II 

15 

i(> 
ir 

IS 

i'.) 

50 
51 
52 
5:1 
51 

55 
:>(( 

57 
5H 
5» 

QO 


36^ 


IngM. 


401  8578 
^02    0X73 

402  3i()7 

401  5460 

402  7753 

403  004^ 
403  2336 
403  4626 
403  6916 

403  9205 

404  1493 
404  37X0 
404  6067 

404  X352 

405  0637 


Diff.  1". 


405 
405 
405 
405 
40(1 

406 
406 
406 

407 
407 

407 
407 
408 
40X 
40X 

408 
408 
409 
409 
409 

409 
410 
410 
410 
410 

410 

4" 
411 

4" 
411 

412 

412 

412 

412 

4«3 

413  2392 

413  4649 

413  6905 

413  9161 

414  1416 

414  3670 
414  5924 

414  8176 

415  0420 
415  2680 


2921 
5205 
748X 
9769 
2051 

433' 
6()ii 

8X89 

116X 

3445 

5721 

7997 

0272 

1547 
4820 

7093 

93''5 
1636 
3907 
6177 

8446 

0714 
2981 
5248 
7S«4 

9780 
2044 
430X 
6571 
8833 
1095 

3356 
5616 

7«75 
0134 


1.415  4930   37.50 


38.26 

3X.24 
3X.23 
38.22 
38.20 

38.19 
38. 18 
38., 7 
38.15 
3X.14 

3>*-i3 
38. 12 
3X.10 
38.09 
38.08 

38.06 
38.05 
3X.03 
38.02 
38.01 

38.00 

37-99 
37-97 
37.96 

37.95 

37-94 
37.92 

37-91 
37..  o 

37-89 

37-87 
37.86 

37-85 
37.84 
37.82 

37.81 
37-80 
37-78 

37-77 
37.76 

37-75 
37-74 
37.72 

37-7« 
37.70 

37.69 
37.68 
37.66 

37-65 
37.64 

37.63 
37.61 
37.60 
37-59 
37-58 

37-56 
37-55 
37-54 
37-53 
37-5< 


37= 


loK  M. 


4«5 

4'5 
415 
416 
416 

416 
416 
4'7 
4'7 

4«7 

4«7 
4'7 
41X 
4.x 
41X 

418 
419 
419 
419 
419 


4930 

7180 

9429 
167X 

39^5 

6172 

84  >  9 
0664 
2909 
5'53 
7:96 
9639 
18X1 
4122 
6362 

X602 
0841 
3079 
53'7 
7554 


419  9790 

420  2026 
420  4260 
420  6494 

420  8728 

421  0960 
421  3192 
421  5423 
421  7654 
421  9884 


422 
422 
422 
422 
423 

423 
423 

4-3 
423 
424 

414 
424 

4^4 
425 
4^5 

425 
4^5 
4^5 
426 

426 

426 
426 
427 
427 
4*7 

427 
4i7 
42X 
428 
428 

428 


2113 

434« 
6569 
X796 

1022 

3248 

5473 
7697 
9920 
2143 

4365 
65X6 
8X07 
1027 
3246 

5465 
7683 
9900 
21 17 
4333 

6548 
8762 
0976 
3189 
5402 

7613 
9824 
2035 
4244 
6453 
8662 


1)1  ft.  1". 

37-50 
37-49 

3747 
37.46 

37-45 

37-44 
37-43 
37-4' 
37.40 

37-39 

37-3^ 
37.37 
37-36 
37-35 
37-33 

37-3* 

3;-3' 

37-30 
37-29 

37-17 

37.26 

37-25 
37.24 

37-23 
37.22 

37.20 

37-19 
37-18 

37-17 
37-16 

i   37-15 
37-13 

37.12 

37-11 

37-10 

37.09 

37.08 
37.06 

37-05 
37.04 

37.03 
37.02 
37.01 
36.99 
36. 9X 

36.97 
36.96 

I    36-95 
!  36.94 

i  36-92 

I  36-91 
36.90 
1  36.89 
I  36.8;! 
I  36.87 

36.86 
36.85 
36.83 
36-82 
36.X1 

36.80 
av3 


38= 


loK  M. 


Diff.  1" 


1.428 
.429 
.429 
.429 
.429 

1.429 
.430 
.430 
.430 
.430 

I.431 

•431 

-431 

-431 

•43' 
1.432 

•432 
.432 
.432 
•433 

1-433 
•43  3 
-43  3 
•433 
•434 

1-434 

434 

4.^ 
435 

435 

1-435 
-435 
•435 
-436 
-436 

1.436 

-436 
.436 

■437 
•437 

1-437 
-437 
.438 

•438 
.438 

1-438 
438 
439 
439 
439 

1-439 

440 

440 

440 

44° 
1.440 
-44' 
•441 
•441 
•44' 

1.44!   9943 


8662 
0X69 

3076 
5281 
7488 

9693 
1X97 
4101 

6304 
X506 

0708 
2909 
5  1 09 
730X 
9507 

1705 
3903 
6100 
X296 
0491 

26X6 
4X8 1 

7074 
9267 

'459 

3651 
5X42 
So  3  2 
0221 
2410 

4598 
67X6 

^97  3 
1159 

3345 

5530 

7714 
9X9X 
20X1 
4263 

6445 

X626 
0X06 
29S6 
5165 

7  344 
9522 
i6q9 

3^75 
6051 

8226 
0401 

2575 
4748 
6921 

9093 
1264 

3436 
5605 

7774 


36.80 
36.79 
36-78 
36-77 
36-75 

36-74 
36-73 
36.72 
36.71 
36-70 

36.69 
36. 6X 
36.66 
3"-6,- 
36.64 

36.63 
36.62 
36.6 1 
36.60 
36-59 

56-57 
36-56 
3 ''-5  5 
36-54 
36-53 

36-52 
36.51 

36-50 
36-49 
36.48 

36-47 
36.46 
36.44 
36.43 
36.42 

36.41 
36.40 
36-39 
36-38 
36-37 

36.36 
36-35 
36-34 
36.32 
36.31 

36.30 
36.29 
36.28 

36-27 
36.26 

36.25 
3(1.24 
36.23 
36.22 
36.20 

36-19 
36.18 
36.17 
36.16 
36.15 

36.14 


39= 


'-447 
447 
447 
448 
448 

1.448 
-448 
-448 
-449 
•449 

1.449 
449 

■1-V9 
450 
.450 

1.450 
•450 
451 
-451 
.451 

1.451 

•451 
•452 
.452 
•452 

I  452 
.4,-2 
•453 
■453 
•453 

1^453 
•454 
•454 
■454 
•454 


loK  M. 

'•44'  9943 
.442  21 1 1 
.442  .1279 
.442  6446 
.442  8612 

1.443  0778 
•443  2943 
•443  5 '07 
■443  727' 
•443  9434 

1.444  1597   I 
-444  3758 
-444   5920 
.444  8cXo 
•445   0240   i 

1-445  2400   , 

•445  4558    I 

-445  671"   I 

.445  8874 

.446  1031 

1.446    3187    ; 

•446  5343   ; 

.446  749  X    ' 

.446  9652 

•447  1806   I 


Diff.  1". 


3959 

(1112 
8263 

0415 
2565 

4715 
6S65 

9014 

1162 
3309 

5456 

7603 

9749 
1X94 
4038 

61X2 
8325 
0468 
2610 

4752 

6X93 
9033 
1173 
3312 
5450 

7588 
9725 
1X62 
3998 
6134 

X269 
0403 

2537 
4670 
6802 


36.14 

36.13 
36.12 
36.1 1 
36.10 

36.09 
36.08 
36.07 
36.06 
36.05 

36.04 
36.03 
36.02 
36.00 
3  5^99 
35.98 

3  5-97 
3  5-';6 

35-95 
35-94 

35  93 
55  92 
5-91 
3  5  90 
35.89 

35-88 
35-87 
3  5  ^'6 
35-85 
35-84 

3583 
35-82 
35.81 
35.80 
35-79 

35-78 

35-77 
3v76 
35-75 
35-74 

35^73 
35-72 
35-7' 
35^70 
3569 
3  5-68 
35-67 
35-66 
3565 
35.64 

35.63 
35.62 
35.61 
35-60 
35-59 

35-58 
35-57 
35-56 
35-55 
35-54 


1-454  8934  I   35.53 


TABLE  VI. 

For  finding  the  True  Anonisily  or  tin-  Tinip  from  the  Perihelion  in  a  Paralxjlic  <)rl)it. 


.1 


1 

40 

° 

41 

0 

l<'i. 

42 

M. 

0 

IMff.  1". 
34-41 

43 

0 

loK  M. 

DIff.  1". 



35-53 

1..K  M. 
1.467    5781 

DIff.  I".. 

I..R  M. 

'•492   3597 

Diff.  1", 

0' 

1.454    8934 

34^95 

1.480 

0627 



33-9' 

1 

•45  5    «°''5 

35^52 

.467   7879 

34^94 

.480 

2691 

3440 

.492 

5631 

33.90 

2 

■455    3*')(' 

35-51 

.467   9976 

34^93 

.480 

4755 

34.40 

.492 

766,- 

3389 

3 

•45?   5326 

35-5° 

.46  S    2071 

34-92 

.4>'o 

6819 

34-39 

.492 

969N 

33-88 

4 

•455   7456 

35-49 

.468   4ib6 

34-9' 

.^8o 

8882 

34.38 

•493 

'73' 

33-87 

3 

'•455   9585 

35.4« 

1.468   6261 

34.90 

1.481 

0944 

34^  ^7 

'•493 

3764 

33-87 

0 

.456   1713 

35-47 

.468    83  5  5 

34.90 

.48, 

3006 

34^36 

•493 

5796 

33.Sf, 

7 

■45'^  3SA1 
•456   59"** 

35-4<' 

.469   0448 

3489 

.481 

5068 

34^35 

•493 

7827 

3  3-><5 

8 

35-45 

.469    2541 

34.88 

.481 

7129 

34^U 

•493 

9858 

3)-"i4 

0 

.456  >lo.;4 

35-44 

.469   4634 

34^87 

.481 

9189 

34-33 

•494 

1888 

33-83 

1     10 

1.457  0220 

3543 

1.469   6725 

34.86 

1.482 

1249 

34-33 

'494 

39 '8 

33-83 

11 

•457  *34<' 

35-42 

.469   8S 17 

34.S5 

.482 

3308 

34-32 

•494 

5948 

33.82 

12 

•457  U70 
•457  6595 

35^4« 

.470  0907 

34.84 

.482 

5367 

34-3' 

•494 

7977 

33-81 

]» 

3  5-4^ 

.4-70  2998 

34-83 

.482 

7425 

34-30 

■495 

0005 

33-80 

M 

•457   87' « 

35-39 

.470   5087 

34-82 

.482 

9483 

34-29 

•495 

2033 

33-79 

13 

1.45X  0841 

35-38 

1.470  7176 

34.81 

1..V83 

1540 

34.28 

'•495 

4061 

'>3-79 

:     10 

.458   2964 

35-37 

.470  9265 

34.80 

•483 

3  597 

34.28 

•495 

608S 

33--8 

17 

.458   50S6 

35-3*' 

■47'    1353 

34--^9 

■4'' 3 

5653 

34-27 

•495 

8114 

33-77 

18 

.45S  7207 

35-35 

-47'    3440 

34-79 

■48' 

7709 

34.26 

.496 

0140 

33-76 

M> 

.458  9328 

3534 

-47'    5527 

34-78 

.483 

9764 

34-25 

.496 

2166 

33-75 

20 

1.459   '44« 

3533 

1.471    7613 

34-77 

1.484 

1819 

34.24 

1.496 

i"'l 

33  75 

21 

•459   3567 

35-3* 

.471    9699 

34.76 

-484   3873 

3423 

.496 

6216 

33-74 

22 

•459   5'''**' 

353« 

■472   1784 

34-75 

-484 

592- 

34-22 

.496 

8240 

33-73 

2;» 

•459  71^05 

35^3o 

-472   3869 

34-74 

-484  7980 

34-22 

•497 

0264 

33-72 

24 

•459  992i 

35^29 

-472   595  3 

34-7  3 

-485 

0033 

34.21 

•497 

2287 

33-7' 

23 

1,460  1040 

35.28 

1-472  8037 

34-73 

1.485 

2085 

34-20 

1.497 

4310 

33-71 

20 

.460  4156 

35^27 

.473   0120 

3472 

•485 

4'37 

34- '9 

•497 

6332 

33-70 

27 

.460  6272 

35^*6 

-473   2203 

34-7' 

-485 

6188 

34- '8 

•497 

8354 

33.09 

28 

.460  8388 

35-15 

-473   4285 

34-70 

•485 

8239 

34-'7 

•498 

0370 

3r(-S 

1    2» 

1 

.461   0503 

35-24 

.473   6366 

^4.69 

.486 

02S9 

34.16 

.498 

2396 

3  3.(,,< 

1   ao 

1.461    2617 

35-23 

1.473   8447 

34.68 

1.486 

2338 

34.16 

1.498 

44' 7 

33-67 

!    :ii 

.461   A731 
.461   6844 

35-23 

.4-4  0527 

34^''7 

.486 

4388 

34  '5 

•498   643- 

33-66 

1    :{2 

35-22 

.474  2607 

34.66 

.486 

6436 

34' 4 

•498 

8456 

33-65 

j     Xi 

.461    8957 

35-2« 

.474  4686 

34-65 

.486 

8484 

34-13 

■499 

0475 

33-''3 

34 

.462   1069 

35-20 

•474  6765 

34.64 

•487 

0532 

34.12 

■499 

2494 

33-6+ 

35 

1.462   3180 

35-19 

1.474  8843 

34-63 

'-487 

2579 

34.12 

1.499 

4512 

33^63 

1     30 

.462   5291 

35^i8 

•475   0921 

34.62 

.487 

4626 

34.11 

-499 

6530 

33-62 

1     37 

.462  7401 

35-17 

•475   2998 

34.61 

-487 

6672 

34.10 

-499 

8547 

33.62 

j     38 

.462   9511 

35.16 

-475   5075 

34.61 

.487 

8718 

34.09 

.500 

0563 

33-61 

3» 

.463    1620 

35-'5 

•475   7«5' 

34.60 

.488 

0763 

34.C8 

.500 

2580 

33.60 

i      lO 

1.463   3729 

35-«4 

1.475   9227 

34-59 

1.488 

2807 

34-07 

1.500 

4595 

33-59 

1     *» 

.463   5837 

35-13 

.47I''   1302 

34-58 

.488 

4852 

34-°7 

.500 

6611 

3  5-5|; 

1     ■»*■« 

•46:   7944 

35.12 

.476   3376 

3457 

.488 

6895 

34.06 

.500 

8625 

33-58 

:  »3 

,464  0051 

35.11 

.476   5450 

34-56 

.488   8939 

34.05 

.501 

0640 

33-57 

44 

.464  2158 

35.10 

-476  7524 

34-55 

.489  0981 

3404 

.501 

2654 

33-56 

43 

1.464  ^263 
.464  6369 

35.09 

1.476  9596 

34-54 

1.489 

3023 

34-03 

1.501 

ll'o^ 

33-55 

40 

35.08 

.477   1669 

34-54 

.489 

5065 

34.02 

.501 

6680 

33-55 

47 

.464  8473 

35-07 

•477   374' 

34-53 

-489 

7106 

34.02 

.501 

8693 

33-54 

48 

.465  0577 

35.06 

.477   58'2 

34-52 

-489 

9'47 

34.01 

.502 

0705 

33-53 

4U 

.465  1681 

3S-°5 

•477  7883 

34-5' 

-490 

1187 

34.00 

.502 

2716 

3352 

50 

1.465  4784 
.465   6886 

35-04 

'•4-'7  9953 

34.50 

1.490 

3127 

33-99 

33.98 

1.502 

i''^l 

3351 

31 

35^04 

.478   2023 

34-49 

-490 

5266 

.502 

^7  3,^ 

33-5" 

52 

.465   8988 

35-03 

.478  4092 

3448 

.490 

7305 

33-97 

-502 

8748 

33-5° 

33 

.466    1090 

35-02 

.478  6161 

34-47 

.490 

9343 

3396 

.503 

0758 

33-49 

34 

.466  3190 

35-01 

.478   8229 

34.46 

•49' 

1381 

33-95 

.503 

1767 

33-48 

35 

1.466   5290 

35.00 

1.479  0197 

34.46 

1.491 

3418 

33-95 

1.503 

ml 

33.48 

50 

.466  7390 

34-99 
34-98 

•479  2364 

34-45 

-49' 

5455 

33  94 

.503 

33-47 

57 

.466  9489 

-479  4-^30 
-479  649'' 

34-44 

-49' 

749' 

3  3-93 

.503 

8792 

33.46 

'     58 

.467   1587 

34-97 

34-43 

-49' 

9527 

3392 

.504 

0800 

33-45 

I     50 

.467  3685 

34.96 

•479  8562 

34.42 

.492 

1562 

339« 

.504 

a8o7 

33-44 

00 

1.467   578Z 

3495 

1.480  0617 

34-41 

1.492 

3597 

35-9« 

1.504 

48.3 

33-44 

576 


Ixdif  Orbit. 


TABLE  VI. 

For  finding  the  True  Anomaly  or  tiie  Time  from  tiie  Perihelion  in  a  Paralwlic  Orhit. 


43^ 

M.    j 

Diir.  1". 

3597 

33-91 

S''3> 

r,.')o 

766  q 

r.^') 

96.>S 

yyXi 

1731 

33-X7 

37<''4 

33''7 

5796 

33^'' 

7X17 

33.!*^ 

,,s,x 

3>'M 

iXXS 

33-^3 

3ViX 

33-^3 

594X 

33.X2 

797-^ 

33.S1 

000; 

33.80 

2033 

3379 

4061 

^3'79 

608S 

3  3-7X 

i!li4 

3  3- '7 

2166 

33-7'' 

33-75 

4191 

33  75 

6216 

33-74 

8240 

3  3-"  3 

0264 

33-7  = 

2287 

33-7« 

4310 

33-71 

f'33z 

33"3 

i*3>4 

33.(19 

03  7(1 

33. hS 

2396 

33.(,S 

44>7 

33.6- 

''437 

33-''(' 

8450 

3)-<'5 

047  i 

33-"5 

2494 

33-''4 

4511 

33-'^3 

6530 

33.1.2 

8547 

3V(u 

05 'n 

33.6, 

2580 

33.1-0 

4i9S 

33->9 

6611 

3^-^** 

862i 

33^'* 

0640 

33-57 

2654 

33-51' 

¥)V 

33-55 

67 

33-55 

80 

33-';5 

9  3 

3'v^+ 

05 

33-53 

lb 

3  5-5- 

i7 

33=;' 

^0 

33-5' 

4X 

33-50 

5X 

33-49 

67 

33-4-^ 

76 

33.48 

84 

3  3-47 

91 

33.46 

00 

33-45 

07 

33-44 

'3 

'  33-44 

r. 


O' 

I 
•i 
» 
1 

5 

A 

7 
H 
U 

10 
II 
Vi 

1:) 
II 

15 
10 
17 
IN 
H) 

W 
•il 
'i'i 
•Z\l 
•il 

!i5 
'Hi 
•Z7 

'iH 
'i9 

30 
31 
M 
33 
31 

35 
3U 
37 

3N 
3U 

40 
II 
i'i 
43 
41 

45 
411 
47 

4N 
41) 

5(> 
51 
5,4 
53 
51 

55 
5(1 
57 

58 
5U 

no 


44 

0 

1..K  M. 

niff.  1". 

504  4813 

33-44 

i;o4  6X19 

33-43 

504  8825 

33-4» 

505  0830 

33-42 

505  2835 

33-41 

505  4839 

33-40 

505  6843 

33-39 

505  884(1 

33-39 

506  0849 

33-3« 

506  2852 

33-37 

506  A854 
506  68^1; 

33-36 

33-36 

506  88i;() 

33-35 

507  0857 

33-34 

507  2857 

33-33 

507  4^57 

33-33 

i;o7  6856 

33-32 

507  8855 

33-3' 

508  o8i;3 

33-30 

508  -^851 

33.29 

508  4849 

33-29 

508  6846 

33.28 

508  8843 

33-27 

509  0839 

33-27 

509  2835 

33.26 

509  4830 

33-25 

509  6825 

33-24 

509  8819 

33-24 

510  0813 

33-23 

510  2807 

33.22 

510  4800 

33.21 

510  6792 

33-21 

510  8785 

33.20 

511  0776 

33-»? 
33.18 

511  2768 

5'«  4759 

33.18 

511  6749 

33-17 

511  8739 

33-16 

512  0729 

33-»5 

512  2718 

33->S 

512  4707 

33-'4 

512  6695 

33-«3 

i;i2  8683 

33-13 

513  0670 

33  12 

5«3  2657 

33-11 

5' 3  4644 

33.11 

513  6630 

33.10 

5'3  J"H5 

33.09 

514  0601 

33.08 

514  2586 

33.07 

514  4S70 

33.07 

5'4  6554 

33.06 

5'4  'S-37 

33-05 

i;!';  0520 

33-05 

5«5  1503 

33.04 

5»5  44«5 

33-04 

515  6467 

33-03 

5«5  i<449 

33-02 

516  0430 

33.01 

516  2410 

33.01 

516  4390 

33.00 

45 

° 

l..g  M. 

DIPT.  1". 

5.6 

4390 

33.00 

516 

6370 

32.99 

516 

8349 

32.98 

517 

0328 

3298 

517 

2306 

32.97 

517 

4284 

32.96 

517 

6262 

32.96 

5'7- 

.8239 

32-95 

518 

0216 

3294 

518 

2192 

32-93 

518 

.J  1 68 

6143 

32.93 

,18 

32,92 

S.8 

8118 

32.91 

5'9 

0093 

32.91 

5«9 

2067 

32.90 

5«9 

4041 

32.89 

519 

6014 

32.89 

519 

7987 

32.88 

519 

9960 

32.87 

520 

1932 

32.86 

520 

3904 

32.86 

S20 

5875 

32.85 

520 

7X46 

32.84 

S20 

9816 

32.84 

521 

1786 

32.83 

521 

3756 

31.82 

521 

5725 

32.82 

521 

7694 

32.8. 

521 

9662 

32.S0 

522 

1630 

32.80 

522 

3598 

32-79 

522 

55<'5 

32-78 

522 

7531 

32.78 

522 

9498 

32.78 

523 

1464 

32-77 

5*3 

3429 

32-76 

523 

5  394 

32-75 

523 

7359 

32-74 

523 

9323 

32-73 

524 

1287 

32-73 

524 

3251 

32.72 

524 

5214 

32.71 

524 

7176 

32.71 

524 

9138 

32.70 

525 

1 100 

32.70 

525 

3062 

32.69 

525 

5023 

32.68 

525 

6983 

32.67 

525 

8944 

32-67 

52b 

0903 

32.66 

,26 

2863 

32.65 

^26 

4822 
6780 

32.64 

526 

32.64 

,26 

8739 

32.63 

527 

0696 

32.62 

527 

2654 

32.62 

527 
527 

461 1 

6567 

32.61 
32.61 

527 

8524 

32.60 

528 

0479 

32.60 

528 

2435 

32.59 

46= 


log  M. 


Dlff.  1". 


28 

2435 

28 

4390 

28 

6344 

28 

8299 

29 

0252 

29 

2206 

29 

4159 

29 

61 12 

29 

806J. 
0010 

30 

30 

1967 

30 

3918 

30 

5869 

30 

7819 

30 

9769 

31 

1719 

3« 

3668 

If 

5616 

31 

7565 

3' 

9513 

32 

1460 

32 

34°7 

32 

5354 

32 

7300 

32 

9246 

33 
33 
33 
33 
33 

34 
34 
34 
34 
34 

35 
35 
5  5 
35 
35 

36 
36 
36 
36 

37 

3' 
37 
37 
37 

37 
38 
38 

^i 

38 

38 
39 
39 
39 
39 

539 


1192 

3'37 

50X2 
7027 
8971 

0914 
285X 
4801 

67.n 
8685 

0627 

2568 
4509 
6450 

8  390 

0330 
2270 
4209 
614X 
8o86 

0024 
1962 
3X99 
5836 

7772 

9708 
1644 
3579 
55'4 
7449 

9383 
1317 

3250 

5183 
71 16 

9048 


32-59 
32.58 
32.57 
32.57 
32.56 

32.55 
3255 
32.54 
32.53 
32.53 

32.5a 

32.5" 
32.51 

32.50 
32.49 

32.49 

32.48 
32.48 
32-47 
32.46 

32.46 
32.45 
32.44 
3244 
32-43 

32-43 
32.42 
32.42 
32.41 
32.40 

32.39 

32-39 
32-38 
32-37 
52.37 

32-36 
32-35 
32-35 
32-34 
32-33 

32-33 
32-32 
32.32 
32.3' 

32.30 

32.30 
32.29 
32.28 
32.28 
32.27 

32.26 
32.26 
32.25 
32-25 
32.24 

32.23 
32.23 
32.22 
32.21 
32,21 

32.20 


47c 


)«K  M. 


5  39  9048 
540  0980 
540  2912 
540  4843 
540  6774 

540  8705 

541  0635 
541  2564 
54 «  4494 
54'  6423 

541  8352 

542  0280 
542  2208 

542  4135 
542  6063 

542  7989 

542  99 "6 

543  18^2 
543  3768 
543  5693 

543  7618 

543  9543 

544  '467 
544  3391 
544  53 '5 

544  7238 

544  9161 

545  1083 
545  3005 
545  4927 
545  6849 

545  8770 

546  0690 
546  7.6 1 1 
546  4531 

546  6450 

546  8370 

547  0289 
547  2207 
547  4125 
547  6043 
547  7961 

547  9878 

548  1795 
548  37«i 

548  5627 
548  7543 

548  9458 

549  '373 
549  328X 

549  5202 
549  7"6 

549  9030 

550  °943 
550  2856 

550  A769 
550  6681 

550  8593 
55'  0504 
55'   2416 

551  4326  I   31.85 


Dlff.  1". 

2. 20 
2.20 
2.19 
2.18 
2.18 

2.17 
2.17 
2.16 
2.15 
2.15 

2.14 

2.14 

2.13 

2.12 
2. 1  I 

2.1  I 
2.10 
2.10 
2.09 
2.09 

2.08 
2.08 
2.07 
2.06 
2.06 

2.05 
2.04 
2.04 
2.03 
2.03 

2.02 

2.02 
2.01 
2.00 
2.00 


.99 
.98 
.98 
.97 
.97 

.96 

.96 
.95 

■94 
-94 

-93 
93 
.92 
.91 
.91 

.90 
.90 
.89 
.^)^ 
.88 

.87 
.87 
.86 
.86 
.85 


■il 


577 


TABLE  VI. 

For  fiiuling  the  True  Anomaly  o>  tlic  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


*♦. 


O' 
I 

a 
» 
1 

3 
G 

7 
8 
U 

lU 
11 
12 

i:i 

14 

13 
IG 
17 
18 
10 

SO 
'^1 
22 
2.-1 
21 

23 
2G 

27 

28 

2a 

:¥) 
:ii 
:i2 

33 
34 

33 

3ii 
37 
38 
30 

40 
41 
42 
43 
44 

43 
4G 

47 
48 
40 

50 
31 
32 
33 
34 

33 
30 
37 

38 
30 

GO 


48 

0 

l..g  M. 

DllT.  1". 

'•55"  43*6 
■55'   6*37 

31.85 

31.84 

•55"    8147 

31.83 

.552  0057 

3'-«3 

.552   19(16 

31.82 

1.552   3876 

31.82 

•552   5784 

31.81 

.552  7693 

31.80 

.552  9601 

31.80 

•553    '5<^8 

3'^79 

1.553  34'6 

3'79 

•553   5323 

31.78 

•553   7230 

31.78 

•553   V'3'' 

3'^77 

•554  '042 

31.76 

•554  4948 

31.76 

•554  4853 
•554  ''758 

3'-75 

3'^75 

•554  8663 

3i^74 

•555  o5''7 

3'^74 

1.555  2472 

3i^73 

•555  4375 

31.73 

•555   6279 

31.72 

.555   81X2 

31.71 

.556  0084 

1.556   1987 

31.70 

.556   388X 

31.70 

.556   5790 

31.69 

.556  7691 

31.68 

.556  9592 

31.68 

'•557   1493 

31.67 

•557  3393 

3'^67 

•557   5293 

31.66 

•557  7«93 

31.66 

.^57  9092 

3'^65 

1.558  0991 

31-65 

.558   2S90 

3'.64 

.558  4788 

31.64 

.558   6686 

31.63 

•558  8584 

31.62 

'•559  0482 

31.62 

•5  59  2379 

31.61 

•559  4275 

31.61 

•559  6'72 

31.60 

.559  8068 

31.60 

1.559  9963 

3«-59 

.560   1859 

3>-59 

.560  3754 

3>^58 

.560  5648 

3'^S7 

.560  7543 

3I-57 

1.560  9437 

31.56 

.561    1331 

31.56 

.561    3224 

3>^5S 

.561   S117 

3«-5S 

.561   7010 

3>54 

1.561   890; 

3>-54 

.562  070A 
.562   2686 

3'-S3 
3i^53 

.562  4578 
.562  6469 

3«-S2 

31.52 

1.562  8360 

3'5' 

49' 


loK  M. 


»I(T.  1 ' 


562  8360 

563  0250 
563  2140 
563  4050 
563  5920 

563  7S09 

563  9698 

564  15S6 

564  347  5 

564  5363 


564 
564 
565 
565 
565 

565 
565 
566 
566 
566 

566 
566 
566 
567 
567 

56- 
567 
567 
568 
568 

568 
'568 
568 
569 
569 

569 
569 
569 
569 

570 


7250 
9' 38 
10^5 
291 1 
4798 

66X4 
8569 
°455 
2340 
4225 
6109 
7993 
9877 
1761 

3644 

5527 
7409 
9291 
"73 

3°55 

4936 
6817 
8698 
0579 
2459 

4338 
6218 
8097 
9976 
1854 


570  3733 

57°  56" 

570  7488 

570  9366 

571  1243 

1.571  3119 

57'  4996 

,571  6872 

571  8748 

572  0623 

572  2499 

572  4373 

572  6248 

572  8123 

572  9997 

573  '870 
573  3743 
573  5616 
573  7489 
573  9362 


1.574  1234 


5» 

5' 

50 
50 
49 

48 
48 

47 
47 
46 

46 
45 
45 
44 
44 

43 
43 
42 
4' 
4> 

40 
40 

39 

39 
38 

38 
37 
37 
36 
36 

35 
35 
34 
34 
33 

33 
32 
32 
3' 

3° 
30 
29 
29 
28 
28 

28 
27 
27 
26 
26 

25 
25 
24 
24 
23 

23 
22 
22 
21 
.21 


31.20 


578 


50' 


lot?  M. 


1.574  I23A 

574  3'°6 
574  4977 
574  6849 

574  8720 


Dtir.  1" 


575 
575 
575 
57  5 
575 

575 
576 
576 
576 
576 

576 

577 
577 
577 
577 

577 
578 

57i! 
578 
578 

578 
578 
579 
579 
579 

579 
579 
580 
580 
580 

580 
580 
581 
58, 
581 

58, 
58, 
581 
582 
582 

582 
582 
582 

583 

583 
584 
584 

584 

584 
584 
584 

585 
58s 


0590 
2461 

433' 
6201 

8070 

9939 
180X 

3677 

5546 

74'4 

9281 

"49 
3016 

4883 

6749  I 

8615  i 
0481  I 

2347  I 
4213  I 
6078  ! 

7942  ' 

9807  I 

1671 

3535  i 

5399  I 

7262 

9125 

0988 

2851 

47'3 

6575 
8436 
0298 

2' 59 
4020 

5880 

7740 
9600 
1460  ' 

33'9  ; 

5179 
7037 

8896  ! 
0754  I 
2612  j 

447°  1 
6327  1 
8184  I 
0041  j 
1898 

3754  i 
5610 

7466 
9321 
1176 

3031 


.20 

.20 

9 

9 

8 

8 
7 
7 
6 

6 

5 
5 
4 
4 
3 

3 
2 

2 
1 
1 

O 

o 

.09 
.09 
.08 

.08 
.07 
.07 
.06 
.06 

.06 

.05 
.04 
.04 
.03 

.03 
.03 

.02 
.02 
.01 

.01 

.00 
.00 

30.99 
30.99 

30.98 
30.98 
30.97 
30.97 

30.96 

30.96 

3<"-95 
30.9s 
30.94 
30.94 

30.94 
30.93 

3"^93 
30.92 
30.92 

30.91 


51 


Ion  M. 


1.585  3031 
585  48X6 
585  6740 
585  859^ 
5X6  0J4S 


586 
586 
5X6 
5X6 
586 

587 
587 
587 
587 
587 

588 
1:88 
5XX 
5X8 
588 

589 
589 
589 
589 

589 

589 

59° 
590 
590 
59° 
590 
591 
59« 
59' 
591 

591 

59' 
C92 

59' 
592 

592 
592 
593 
593 
593 

593 
593 
593 
594 
594 

594  5429 
594  7270 

594  91" 

595  °952 
595  a792 

595  4633 
595  6473 

595  83'2 

596  0151 
596  1990 

596  3829 


2302  I 

4' 5  5  ' 
6008 

7859  i 
97' 3  i 

'565  I 
3417  I 
526X 
7120 
8971 

0821 
2672 
4522 
6372 
8222 

0071 
1920 

3769 
5618 
7466  I 

93'4  j 
1162  I 

3009  I 

4857  i 
6704  I 

8550  I 

0397  I 
2243  I 

4089  ; 

5935  I 

7780  j 
9625  I 
1470  I 
33'5  ] 
5'59  I 

7003 ! 

8847  ! 

0690  j 

2534 

4377 

6219 

8062 
9904 
1746 
3588 


DIff.  1". 

30.91 
30.91 
30.90 
30.00 
30.89 

30.89 
30.89 
30.88 
30.87 
30.87 

30.87 
30.86 
30.86 
30.85 
30.85 

30.8+ 
30.84 
30.83 
30.83 
30.83 

30.82 
30.82 
30.81 
30.81 

30.80 

30.80 
30.79 
30.79 
30.78 
30.78 

30.78 

30^77 
30.77 
30.76 
30.76 

3°^75 
30^75 
3°^75 
30^74 
3°^74 

30.7] 

30^73 
30.72 
30.72 
30.72 

30.71 
30.71 
30.70 
30.70 
30.69 

30.69 
30.68 
30.68 
30.68 
30.67 

30.67 
30.66 
30.66 
30.65 
30.65 

30.65 


bolic  Orbit. 


TABLE  VI. 

For  finding  the  True  Anomaly  or  (he  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


51^ 


M. 


5031  ■ 
48X6 
6740  i 

OJ48  ' 
2302 

4«55 
600S 

7«59  i 
97' 3  I 

1565 
34»7 
5268 
7120 
8971 

0821 

2(171 

4S-* 
6372 
8222 

I  0071 
I  1920 
I  3769 
(  561X 
I  7466 


93J4 
1162 

3009 

4857 
6704 

8550 
0397 
2243 
4089 
5935  \ 
7780  I 
9625  1 
1470  I 
33' 5  1 
S'59  I 

7003  ; 
8847  I 
0690  ! 

1534  ; 
4377  I 

6219  i 

8062 

9904 

1746 

3588 

54»9 
7270 
9111 
0952 
2792 

>5  4633 
)5  H7  3 
,5  8312 
j6  0151 
96  199° 


96   3829 


Diff.  1". 

30.91 
3c..)  I 
30.90 
30.90 
30.89 

30.89 
30.89 
3C.X8 
30.87 
30.87 

I  30.X6 
i    30.86 

1  30'i*> 
I   3°-i'5 

i  30-i'4 
\   30.84 

!  ^°-^ 
I  30.83 
1  30.83 

1  3°-8^ 

I  30.82 

I  30.X1 

.  30.S1 

1  30.80 

30.80 
30.79 
30.79 
30.78 
30.78 

30.7X 

30'77 
30.77 
30.76 
30.76 

3°-7') 
30.75 

I  3°'7S 
i  3°-7+ 
I   3°-74 

;  3°'73 

I  30'73 

!  3°-7^ 

i  3°-7i 

I  30'7i 

i   3°-7' 

-   30.71 

30.70 

30.70 

30.69 

30.69 
30.68 
30.68 
30.68 
30.67 

30.67 
30.66 
30.66 
30.65 
30.65 

30.65 


r. 

52 

0 

53 

0 

54 

0 

55 

0 

loK  M. 

Diflr.  1". 

l"t 

M. 

DllT.  1". 

logM. 

Via.  1". 

logM. 

Diff.l".  ' 

0' 

1.596  3829 

30.65 

1.607 

3703 

30.40 

1.618  2724 

30.17 

1.629  0959 

29.96 

1 

.596  5668 

30.64 

.607 

5527 

30.39 

.618  4534 

30.17 

.629  2757 

29.96   , 

•i 

.596  7506 

30.64 

.607 

7350 

30.39 

618  6344 

30.16 

.629  4554 

29.96 

3 

■59<'  9344 
.597  1182 

30.63 

.607 

9 '74 

30.39 

.618  8153 

30.16 

.629  6351 

29^95 

4 

30.63 

.608 

0-97 

30.38 

.618  9963 

30.16 

.629  8148 

2995   j 

5 

1.597  3020 

30.62 

1.608 

2820 

30.38 

1.619  1772 

30.15 

1.629  9945 

29.95 

0 

•597  4**57 

30.61 

.6c8 

4642 

30.38 

.619  3581 

30.15 

.630  1742 

29.94 

7 

•597  6694 

30.62 

.6c8 

6465 

30-37 

.619  5390 

30,15 

.630  353S 

29.94   . 

H 

•597  ii53' 

30.61 

.608 

8287 

30.37 

.619  7199 

30.14 

,630  5335 

29-94   1 

9 

.598  0368 

30.61 

.609 

0109 

30.36 

.619  9007 

30.14 

,630  7131 

29-93   1 

10 

1.598  2204 

30.60 

1.609 

'93' 

30.36 

1.620  0816 

30.14 

1.630  8927 

29.93   ' 

II 

.598  4040 

30.60 

.609 

3752 

30.36 

.620  2623 

30.13 

.631  0722 

29-93 

1*4 

.598  5876 

30.59 

.609 

5573 

30.35 

.620  4431 

30.13 

.631  2518 

29.92  : 

i:i 

.598  77 1 1 

30.59 

.609 

7394 

30.35 

.620  6239 

30.12 

.631  4313 

29.92   : 

II 

•598  9547 

30^59 

.609 

9215 

30^34 

.620  8046 

30.12 

.631  6108 

29.92 

15 

1.599  >3»a 

30.58 

1.610 

'S^^ 

30.34 

1.620  9853 

30.12 

1.631  7903 

29.91 

la 

•599  31 '7 

30.58 

.610 

2856 

30.34 

.621  1660 

30.11 

.631  9698 

29.91 

17 

•599  505' 

30-57 

.610 

4676 

30.33 

.621  3467 

30,11 

.632  1492 

29.91 

18 

.599  6885 

3°-57 

.610 

6496 

30.33 

.621  5274 

30.11 

.632  3286 

29.90 

l» 

•599  87 '9 

30-57 

.610 

83,5 

30.32 

.621  7080 

30.10 

.632  5081 

29.90 

20 

1.600  0553 

30.56 

1.611 

0135 

30.32 

1.621  8886 

30.10 

1.632  6875 

29.90 

•il 

.600  2387 

30.56 

.611 

'954 

30.32 

.622  0692 

30.10 

,632  8668 

29.89 

•i'Z 

.600  4220 

30^55 

.611 

3773 

30.31 

.622  2497 

30.09 

.633  0462 

29.89 

•a 

.600  6053 

30^55 

.611 

559' 

30.31 

.622  4303 

30.09 

.633  2255 

29.89 

•z\ 

.600  7886 

30^55 

.611 

7410 

30.31 

.622  6108 

30.09 

.633  4048 

29.88  ' 

25 

1.600  9718 

30.54 

1.611 

9228 

30-30 

1.622  7913 

30.08 

1.633  5841 

29.88 

2i( 

.601  1551 

30.54 

.612 

1046 

30.30 

.622  9718 

30.08 

.633  7634 

29.88 

27 

.6oi  3383 

30-53 

.612 

2864 

30.29 

.623  1523 

30.08 

,633  9427 

29.87 

2H 

.601  5214 

30-53 

.612 

4681 

30.29 

.623  3327 

30.07 

.634  1219 

29.87 

29 

.601  7046 

30.52 

.612 

6499 

30.29 

.623  5131 

30.07 

.634  3011 

29.87 

30 

1.601  8877 

30.52 

1.612 

8316 

30.28 

1.623  6935 

30.06 

1.634  4803 

29.86 

31 

.602  0708 

30.52 

.613 

0132 

30.28 

.623  8739 

30.06 

,634  6595 

29.86 

32 

.602  2539 

30.51 

.6,3 

1949 

30.28 

.624  0543 

30.06 

,634  8387 

29.S6 

33 

.602  4370 

30-5' 

.6.3 

3765 

30.27 

.624  2346 

30.05 

,635  0178 

29.86 

3t 

.602  6200 

.30.50 

.613 

5582 

30.27 

.624  4149 

30.05 

.635  1969 

29.85  : 

35 

1.602  8030 
.602  9860 

30.50 

1.613 

7398 

30.26 

1.624  5952 

30.05 

1.635  3760 

29.85 

30 

30.50 

.6.3 

9213 

30.26 

.624  7755 

30.04 

-635  555' 

29.85 

37 

.603  1690 

30.49 

.614 

1029 

30.26 

.624  9557 

30.04 

.635  7342 

29.84 

38 

.603  3519 

30.49 

.6,4 

2844 

30.25 

.625  1360 

30.04 

.635  9132 

29.84 

39 

.603  5348 

30.48 

.614 

4659 

30-25 

.625  3161 

30.03 

.636  0922 

29.84   , 

40 

1.603  7«77 

30.48 

1. 614 

828^ 

30.25 

1.625  4964 
.625  6765 

30.03 

1.636  2713 

29.83 

11 

.603  9005 

30-47 

.614 

30.24 

30.03 

.636  4502 

29.83 

42 

.604  0834 
.604  2662 

30.47 

.6,5 

0103 

10.24 

.625  8567 

30.02 

.636  6292 

29.83 

43 

30.47 

.615 

1917 

30.23 

.626  0368 

30.02 

.636  8082 

29.82 

41 

.604  44  . 

:o.46 

.615 

373' 

30.23 

.626  2169 

30.02 

.636  9871 

29.82 

15 

1.604  ''3'7 

30.46 

1.615 

5545 

30.23 

1.626  3970 

30.01 

1.637  1660 

29.82 

4» 

.604  8145 

30.45 

.615 

7358 

30.22 

.626  S77I 

30.01 

■637  2449 

29.82 

47 

.604  9972 

30.45 

.6,5 

9171 

30.22 

.626  7571 

30.01 

.637  5238 

29.81 

48 

.605  1799 

30.45 

.616 

0984 

30.22 

.626  9372 

30.00 

.637  7027 

29.81 

49 

.605  3626 

30.44 

.616 

2797 

30,21 

.627  1172 

30.00 

.637  8S15 

29.81 

50 

1.605  5452 

3044 

1.616 

4610 
6422 

30,21 

1.627  2972 

30.00 

1.63?  0603 

29.80 

.'il 

.605  7278 

30.43 

.616 

30.20 

.627  4771 

29.99 

.638  2391 

29.80 

:)2 

.605  9104 

30.43 

.616 

8234 

30,20 

.627  6571 

29.99 

.638  4179 

29.80 

■>:i 

.606  0930 

30.43 

.617 

0046 

30.20 

.627  8370 

29.99 

.638  5967 

29.79 

54 

.606  2755 

30.42 

.617 

1858 

30,19 

.628  0169 

29.98 

.638  7754 

29.79 

.■|5 

1.606  4581 

30.41 

1. 617 

3669 

30.19 

1.628  1968 

29.98 

1.638  9542 

19.79 

29.78    1 

50 

.606  6406 

30.41 

.617 

5481 

30.19 
30,18 

.618  3766 

2998 

.639  1329 

57 

.606  8230 

30.41 

.617 

7292 

.628  5565 

29.97 

.639  31 16 

29.78 

58 

.bo7  0055 

30.41 

.617 

9101 

30,18 

.628  7363 

29.97 

.639  4902 

29.78 

59 

.607  1879 

30.40 

.618 

0913 

30.17 

.628  9161 

29.97 

.639  6689 

29^77 

6(1 

1,607  3703 

30.40 

1.618 

2724 

30.17 

1.629  0959 

29.96 

1.639  8475 

29.77 

_  1 

670 

TABLE  VI. 

For  finiiing  tlic  True  Anomaly  or  tlie  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


li 


V. 

56 

0 

57 

3 

58 

0 

59 

0             1 

1 

loK  M. 

Wfr.  1". 
29.77 

l(.(l  M. 

mir.  1". 

)..«  .M. 

Diff.  1". 

loK  M. 

Diff.  1'.  ! 

o 

1.639    8475 

1.650  5336 

29.60 

1.661    1601 

29.44 

1.671   7331 

29.30 

1 

.640   0262 

29.77 

.650  7112 

29.60 

.661    3368 

29.44 

.671  9089 

29.30 

a 

.640    204X 

29.77 

.650  8887 

29.59 

.661    5134 

29-44 

.672  0846 

29.30 

:i 

.640    3X33 

29.76 

.65.    0663 

29.59 

.661    6900 

29.43 

.672  2604 

29.19 

4 

.640    5619 

29.76 

.651   2438 

29.59 

.661   8666 

29.43 

.672  4362 

29.19 

5 

1.640    7405 

29.76 

1. 651   4213 

29.58 

1.662  0432 

29-43 

1.672  6119 

29.29 

» 

.640    9190 

19-75 

.651    5988 

29.58 

.662  2197 

29.43 

.672  7876 

29.29     , 

T 

.641    097  s 

29.75 

.651    7763 
.651   9538 

29.58 

.662   3963 
.662  5728 

29.42 

.672  9634 

29.2X 

8 

.641    2760 

29.75 

29.58 

29.42 

.673   1391 

29.28 

1        ^ 

.641    4545 

29.74 

.652   1312 

»9-57 

.662  7493 

29.42 

.673  3147 

29.18 

!     10 

I.641    6329 

29.74 

1.652   3086 

29-57 

1.662  9258 

29.42 

1.673  4904 

29.28 

I     11 

.641    8114 

29.74 

.652  4861 

i9-57 

.663    1023 
.663  2788 

29.41 

.673  6661 

29.28 

1     1« 

.641    9898 

*9-74 

.652  6635 

*9.57 

29.41 

.673  8417 

29.27 

;    13 

.642    1682 

29.73 

.652  8408 

29.56 

.663  4553 

29.41 

.674  0174 

29.27 

1    »• 

.642    3466 

29.73 

.653   0182 

29.56 

.663  6317 

19.41 

.674  1930 

29.27 

:     15 

1.642    5250 

29.73 

1.653    1956 

29.56 

1.663  8082 

29.40 

1.674  3686 

29.27 

'  V: 

.642    7033 

29.72 

.653    3729 

*9-55 

.663  9846 

29.40 

.674  5442 

29.27 

17 

.642   8Xj6 

29.72 

.653   5502 

i9-S5 

.664   1610 

29.40 

.674  7198 

29.26 

18 

.643  0599 

29.72 

.653   7275 

29-55 

.664  3374 

29.40 

.674  8954 

29.26 

1» 

.643  2382 

29.71 

.653   9048 

*9-55 

.664  5137 

29.39 

.675  0709 

29.26 

20 

1.643  4165 

29.71 

1.654  °*'*' 

29.54 

1.664  6901 

29.39 

1.675  2465 

29.26 

{     ^1 

.643   5948 

29.71 

•654   2593 
.654  4366 

29.54 

.664  8664 
.665  0428 

29-39 

.675  4220 

29.25 

22 

.643   7730 

29.71 

29.54 

29.39 

•675  5975 

29.25 

2:i 

.643  9513 

29.70 

.654  613S 

49-54 

.665  2191 

29.39 
29-38 

.675  7730 

29.25 

21 

.644  129s 

29.70 

.654  7910 

29-53 

.665   3954 

.675  9485 

29.25 

25 

1.644   3077 

29.70 

1.654  9682 

29-53 

1.665   5717 

29.38 

1.676  1240 

29.25 

2(( 

.644  4858 

29.69 

.655    1454 

29-53 

.665   7480 

29.38 

.676  2995 

29.24 

27 

.644  6640 

29.69 

.655    3225 

29-53 

.665  9242 

29.38 

.676  4749 

29.24 

28 

.644  8421 

29.69 

.655   4997 

29.52 

.666   1005 

29-37 

.676  6504 

29.24 

2« 

.645  0203 

29.69 

.655   6768 

29-52 

.666  2767 

29.37 

.676  8258 

29.24 

30 

1.645    '984 

29.68 

••655   8539 

29-52 

1.666  4529 

29.37 

1.677  0012 

29.24 

31 

■64s   3765 

29.68 

.65b   0310 

29.51 

.666  6291 

29-37 

.677   1766 

29.23 

32 

•645  5545 

29.68 

.656   2081 

2951 

.666  8053 

29.36 

.677   3520 

29.23 

1     33 

.645  7326 

29.67 

.656   3852 

29.51 

.666  9815 

29-36 

.677  527.1 
.677  7028 

29.23 

j     3.1- 

.645  9106 

29,67 

.656   5622 

29.51 

.667    1577 

2936 

29.23 

35 

1.646  0886 

29.67 

1.656   7392 

29.50 

1.667   3338 

29.36 

1.677  8781 

29.23 

30 

.646  2666 

29.67 

.656   9163 

29.50 

.667   5100 

29-35 

.678  0535 
.678  2288 

29.22 

37 

.646  A446 
.646  6226 

29.66 

.657  C933 

29.50 

.667  6861 

29.35 

29.22 

38 

29.66 

.657   2703 

29.50 

.667   8622 

29.35 

.678  4041 

29.22 

30 

.646  8005 

29.66 

.657  4472 

29-49 

.668  0383 

29-35 

.678  5794 

29.22 

10 

1.646  9785 

29.65 

1.657   6242 

29.49 

1.668  2144 

29-35 

1.678  7547 

29.22 

41 

.647  1564 

29.65 

.657   8011 

2949 

.668   3904 

29.34 

.678  9300 

29.21 

42 

•647   3343 

29.65 

.657  9781 

29.49 

.668   5665 

29.34 

.679  1053 
.679  2806 

29.21 

43 

.647  5122 

29.65 

.658    1550 

29.48 

.668  7425 

29.34 

29.21 

44 

.647  6900 

29.64 

.658   3318 

29.48 

.668  9185 

29.34 

.679  4558 

29.21 

45 

1.647  8679 

29.64 

1.658   5087 

29.48 

1.669  0945 

29.33 

1.679  6310 

29.20 

4G 

.648  0457 

29.64 

.658   6855 

29.48 

.669  2705 

29.33 

.679  8063 

29.20 

47 

.648  2235 

29.63 

.658   8624 

29.47 

.669  A465 
.669  6225 

29-33 

.679  9815 

29.20 

48 

.648  4013 

29.63 

.659  0393 

29.47 

29-33 

.680  1567 

29.20 

40 

.648   5791 

29.63 

.659  2161 

29-47 

.669  7984 

29.32 

.680  3319 

29.20 

50 

1.648   7569 

29.63 

1.659   3929 

29-47 

1.669  9744 

29.32 

1.680  5070 

29.19 

51 

.648  9346 

39.62 

.659  5697 

29.46 

.670   1503 

29.32 

.680  6822 

29.19 

52 

.649   1123 

29.62 

.659  7465 

29.46 

.670  3262 

29.32 

.680  8574 

29.19 

53 

.649  2901 

29.62 

.659  9232 

29.46 

.670  5021 

29.32 

.681  0325 

29.19 

54 

.649  4677 

29.61 

.660   1000 

29.46 

.670  6780 

29.31 

.681  2076 

29.19 

55 

1.649  6454 

29.61 

1.660  2767 

29-45 

1.670  8539 

29.31 

1.681   3827 

29.18 

5G 

.649   8231 

29.61 

1^°  i"' 
.660  6301 

29.45 

.671   0298 

29.31 

.681   5578 

29.18 

57 

.650  0007 

29.61 

29.45 

.671   2056 

29.31 

.681  7329 

29.18 

58 

.650   1784 

29.60 

.660  8068 

29.45 

.671   3814 

29.30 

.681  9080 

29.1S 

50 

.650  3560 

29.60 

.660  9835 

29.44 

.671   5573 

29.30 

.682  0831 

29.18 

GO 

1.650  5336 

29.60 

1. 661    1601 

29.44 

1.671   7331 

29.30 

1.682  2581 

29-17 

680 


ibolic  Orbit. 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Tiiue  trom  the  IVriliulion  in  a  Parabolic  Orbit. 


59° 

K  M.    1 

Diff.  v.  ; 

7331  i 
9089  ' 
0846 
.  2604 
■  436*  ' 

29.30 
29.30 
29.30 
29.19 
29.19 

'.  6119 
I  7876  ! 
I  9634  ; 

5  »39«  i 
3  3>47  I 

3  4904  ! 

3  6661 

3  84«7 

4  0174 
4  1930 

4  3686 
4  5441 
4  7'9» 

4  8954 

5  0709 

5  2465 
5  4220 

'5  5975 
'5  7730 
'5  9485 
r6  1240 
id   2995  I 

?6  4749 
76  6504  i 

76  8258  1 

77  0012 
77  1766  ! 
77  3520  1 
77  5V4  ' 

77  7028  ! 

77  8781  I 

78  0535  ' 
78  2288  ! 
78  4041  ; 
78  5794  ; 

178  7547 

178  9300 

179  1053 
179  2806 
>79  4558 
)79  6310 
179  8063 

J70  98' 5 
580  1 567 
38o  3319 

680  5070 
58o  6822 

680  8574 

681  0325 
681  2076 

681  3827 
681  5578 
681  7329 

681  9080 

682  0831 

682  2581 


29.19 
29.19 
29.28 
29.18 
29.28 

29.18 
29.18 
29.17 
29.17 
29.27 

29.27 

29.27 

29.16 

t   29.16 

I   29.26 

i  29.16 
19.25 
I  29.25 
I  29--5 
I   49'^5 

!    29.15 

I     29.14 

29.14 

;  29.14 
1   19--4 

i  29.14 
I   29.23 

j  29-23 
i   29-23 

!     29-23 

I  29.13 
i  29.11 
!     29.12 

;  29.11 
29.11 

29.11 
29.11 
19.11 
29.11 
29.21 

29.10 
29.10 
29.10 
29.10 
29.10 

29.19 
29.19 
29.19 
29.19 
29.19 

29.18 
29.18 
29.18 

29. 18 
29.18 

29' '7 


V, 


O' 

I 

'Z 

3 

4 

5 
G 

*r 
4 

8 
U 

lU 
11 
\'Z 
13 
11 

15 
Hi 
17 
18 
10 

20 
21 
22 
23 
24 

25 
2G 
27 
28 
20 

30 
31 
32 
33 
34 

35 
36 
37 

.38 
30 

40 
41 
42 
43 
44 

45 
40 
47 

48 
40 

50 
51 
52 
53 
54 

55 
50 
57 
58 
50 

60 


60^ 


li.g  M.        I   Diff.  1" 


7568 
93'6 
1064 
2812 
4560 

6308 
8c55 
9803 


1.682  2581 
.682  4332 
.682  6082 
.682  7832 
.682  9582 

1.683  133a 
.683  3082 
.683  4832 
.683  6581 
.683  8331 

1.684  0080 
.684  1830 
.684  3579 
.684  5328 
.684  7077 

1.684  8826 
.685  0574 
.685  2323 
.685  4071 
.685  5820 

1.685 
.685 
.686 
.686 
.686 

1.686 
.686 
.686 
.687 
.687 

1.687 
.6S7 
.687 
.688 
.688 

1.6S8 
.688 
.688 
.688 
.689 

1.689 
.689 
.689 
.689 
.689 

1.690 
.690 
.690 
.690 
.690 

1.690 
.691 
.691 
.691 
.691 

1. 691 
.692 
.692 
.692 
.692 

1.692 


29.17 
19.17 
29.17 
29.17 
29.17 

29.16 
29.16 
29.16 
29.16 
29. 1 6 

29.16 
29.1  c 
29.15 
29.15 
29.15 

29.14 
29.14 
29.14 
29.14 
29.14 

29.14 
29.13 
29.13 
29>3 
29- 1 3 

*9-i3 
29.13 
29.12 


1550 

29.12 

3297 

29.12 

5044 

29.12 

6791 

29.12 

8538 

29.11 

0285 

29.11 

20J2 

29.11 

3778 

29.11 

5525 

29.11 

7271 

29.10 

9017 

29.10 

0764 

29.10 

2510 

29.10 

4256 

29.10 

6001 

29.09 

7747 

29.09 

9493 

29.09 

1238 

29.09 

2984 

29.09 

4729 

29.09 

6474 

29.09 

8219 

29.08 

9964 

29.08 

1709 

29.08 

3454 

29.08 

S«99 

29.08 

6943 

29.08 

8688 

29.07 

0432 

29.07 

2176 

29.07 

3920 

29.07 

5664 

29.07 

7408 

29.07 

61^ 


l(>K  M. 


1.692 
.692 

.693 
.693 
.693 

1.693 
.693 
.693 
.694 
.694 

1.694 
.694 
.694 
.695 
.695 

1.695 
.695 
.695 
.695 
.696 

i.6g6 
.696 
.696 
.696 
.696 

1.697 
.697 
.697 
.697 
.697 

1.697 
.698 
.698 
.698 
.698 

1.698 
.699 
.699 
.699 
.699 

1.699 
.699 

.700 
.700 
.700 

1.700 
.700 
.700 
.701 
.701 

1. 701 

.701 
.701 
.701 
.702 

1.702  3174 
.702  4913 
.702  6651 
.702  8389 
.703  0128 

1.703  1866 


7408 
9152 
0896 
2640 
4383 

6127 
7870 
9613 
1356 
3099 

4842 

6585 
8328 
0070 
1813 

3555 
5298 
7040 
8782 
0524 

2266 
4008 
5750 
749" 
9233 
0974 
2716 

4457 
6198 

7939 
9680 
1421 
3162 
4902 
6643 

8383 
0124 
1864 
3604 
5345 
7085 
8824 
0564 
2304 
4044 

5783 
7523 
9262 

ICOI 

2741 

4480 
6219 

7958 
9697 

1435 


Diff.  1". 

29.07 
29.06 
29.06 
29.06 
29.06 

29.06 
29.05 
29.05 
29.05 
29.05 

29.05 
29.04 
29.04 
29.04 
29.04 

29.04 
29.04 
29.04 
29.03 
29.03 

29.03 
29.03 
29.03 

29-03 
29.02 

29.02 
29.02 
29.02 
29.02 
29.02 

29.02 
29.01 
29.01 
29.01 
29.01 

29.01 
29.01 

29.00 
29.00 
29.00 

29.00 
29.00 
29.00 
29.00 
28.99 

28.99 
18.99 
28.99 
28.99 
28.99 

28.98 
28.98 
28.98 
28.98 
28.98 

28.98 
28.98 
28.97 
28.97 
28.97 

23.97 
581 


62= 


loK  .M. 


5293 

7029 

7  *'  5 

0501 
2237 

3972 
5708 
7444 
9«79 
0914 

2650 

4385 
61  20 

7855 
959° 


Diff.  I" 


1.703  1866 

703  3604  ! 

703  53-f2 

703  7080 

703  .S818  I 

704  0556  i 
704  2293  I 
704  4031 
704  576S  i 
704  7506 

I 

704  9243  I 

705  0981  j 
705  2718  , 

705  4455  I 

705  6192  I 

705  7929  ; 

705  9666  , 

706  1402 
706  3139  I 
706  4875 

706  6612  I 

706  8348  j 

707  0085  I 
707  1 821  i 
707  3557  I 


707 
707 
707 
708 
708 

708 
708 
708 
708 
709 

709 
709 
709 
709 
709 

710  1325 
710  3060 
710  4794 
710  6529 
710  8263 

710  9998 
7U  1732 

711  3467 
711  5201 
711  6935 

711  8669 

712  0403 
712  21 37 
712  3871 
712  5605 

7«2  7339 

712  9072 

713  0806 
713  2539 
7>3  4273 
713  6006 


28.97 
28.97 
18.97 
28.97 
28.96 

28.96 
28.96 
28.96 
28.96 
28.96 

28.96 
28.95 
28.95 
28.95 
28.95 

28.95 
28.95 
28.95 
28.94 

28.94 

28.94 
28.94 
28.94 
28.94 
28.94 

28.93 
28.9- 
28.93 
28.93 
28.93 
28.93 
2893 
28. 92 
28.92 
28.92 

28.92 
28.92 
28.92 
28.92 
28.92 

28.91 
28.91 
28.91 
28.91 
28.91 

28.91 
28.91 
28.90 
28.90 
28.90 

28.90 
28.90 
28.90 
28.90 
28.90 

28.90 
28.89 
28.89 
28.89 
28.89 

28.89 


63= 


log  .M. 


6006 

7739 

9473 
1206  I 

2939  i 

4672  ' 

6405 

8138 

9870 

1603 

3336  i 

5068 

6801 

8533 
0266 

1998 

3730 
5462 

7'94 
8926  ; 

0658  i 
2390  ; 
4122  i 

5853 : 
7585  : 

9317  \ 
1048 
2780 1 

45"  ! 

6242  I 

7974  ' 
9705 
1436  . 

3«67  i 
4898  I 

6629 
8360 
0090 
1821 

3552  i 
5282  . 
7o>3  i 
8743 
0474  . 
2204  : 

721  3934 
721  5665 

721  7395 

721  9125 

722  0855 


7 
8 
8 
8 
8 

8 
8 
9 
9 
9 

9 
9 

720 
710 
720 

720 
720 
720 
721 
721 


Mff  1". 

28.89 
28.89 
28. 89 
28. 88 
28.88 

28.88 
28.88 
28. 88 
28.88 
28.88 

28.S8 
28. 88 
28.87 
28.87 
28.87 

28.87 
28.87 
28.87 
28.87 
28. 87 

28.86 
28.86 
28.86 
28.86 
28.S6 

28.86 
28.86 
28. 86 
28. 86 
28.85 

28.85 
28.85 
28.85 
28.85 
28.85 

28. 85 
28.85 
28.85 
28.84 
28.84 

28.84 
28.84 
28.84 
28.84 
28.84 

28.84 
28.84 
28.84 
28.83 
28.83 


722  2585 

28.83 

722  4315 

28  83   1 

722  6044 

28.83   1 

722  7774 

28.83 

,22  9504 

28.83 

723  1233 

28.X3 

723  2963 

28.83 

723  4693 

28.82 

723  6422 

28.82   i 

723  8151 

28.82   1 

723  9881 

28.82 

TABLE  VI. 

For  findinp;  tlic  True  Anoniiily  or  tlic  Tiiiie  from  llie  IVrilii-lioii  in  a  Parabolic  Orbit. 


'/ 


1    V. 

i 

1      0' 

64 

0 

65 

0 

66 

0 

67 

0 

logM. 

Via.  1". 
28.82 

log 

'•'34 

M, 

3539 

I)lff.  1". 

28.77 

log 

M. 

KitT.  I". 

28.73 

\ov 

[M. 

IMIT.  1". 

1.723 

9881 

1.744 

7031 

'•755 

0405 

28.70 

!     1 

•71+ 

1610 

28.82 

•734 

5265 

28.77 

•744 

8755 

28.73 

•755 

2127 

28.70 

a 

.724 

3339 

28.82 

•^34 

6y.;l 

28.77 

•745 

0479 

28.73 

•755 

3849 

28.70 

1       3 

.72+ 

<;o68 

28.82 

•734 

8718 

28.77 

•745 

2202 

28.73 

•755 

5  57' 

28.70 

4 

.724 

6798 

28.82 

•735 

0444 

28.77 

•745 

3926 

28.73 

•755 

7293 

2X.70 

5 

1.724 

8527 

28.82 

'•735 

2169 

28.76 

'•745 

5650 

28.73 

'•755 

9015 

28.70 

0 

■7-S 

0256 

28.82 

•735 

3895 

28.76 

•74; 

7373 

28.73 

.756 

0737 

28.70 

;      7 

•715 

1984 

28.81 

•735 

5621 

28.76 

•745 

9097 

28.73 

.756 

2459 

28.70 

« 

•725 

37'3 

28.81 

•735 

7347 

28.76 

.746 

0820 

28.72 

.756 

4181 

28,70 

I       0 

•725 

544* 

28.81 

•735 

9073 

28.76 

.746 

2544 

28.72 

.756 

5903 

28.70 

!    10 

1.725 

7171 

28.81 

1.736 

0798 

28.76 

1.746 

4267 

28. 72 

1.756 

7625 

28. 70 

II 

•  72'; 

8900 

28.81 

•  736 

2524 

28.76 

,746 

599' 

2S.72 

.756 

9347 

28.70 

vz 

.726 

0628 

28.81 

.736 

425° 

28.76 

.746 

77'4 

28. 72 

•757 

1C69 

28.70 

1     13 

.726 

2357 

28.81 

.736 

5975 

28.76 

.746 

9437 

28.72 

•757 

2791 

28.70 

'     14 

.726  40X5 

28.81 

.736 

7701 

28.76 

•747 

Il6i 

28.72 

•757 

45'3 

28.70 

Kt 

1.726 

5814 

28.81 

1.736  9426 

28.76 

'•747 

2884 

28.72 

'•757 

6235 

28.70 

1(( 

.726 

7542 

28.81 

•737 

1152 

28.76 

•747 

4607 

28.72 

•757 

7957 

28.70 

17 

.726 

9270 

28.81 

•737 

2877 

28.76 

•747 

6330 

28.72 

•757 

9679 

28.70 

IH 

•727 

0999 

28.80 

•737 

4602 

2S.76 

•747 

8054 

28.72 

•758 

1401 

28.70 

l« 

•727 

2727 

28.80 

•737 

6328 

28.75 

•747 

9777 

28.72 

.758 

3'23 

28.70 

«0 

1.727 

6183 

28.80 

'•737 

8053 

28.75 

1.748 

1500 

28.72 

..758 

4844 
6566 

28.70 

21 

•727 

28. 80 

•737 

9778 

28.75 

.748 

3223 
4946 

28.72 

.758 

28.70 

aa 

•727 

79" 

28.80 

.738 

'5°3 

28.75 

•748 

28.72 

.758 

8288 

28.70 

23 

•727 

9639 

28.80 

.738 

3228 

28.75 

.748   6669 

28.72 

•759 

0010 

28.70 

24 

.728 

1367 

28.80 

.738 

4953 

28^75 

.748 

8392 

28.72 

•759 

1731 

28.70 

25 

1.728 

309- 

28.80 

1.738 

6679 

28.75 

'•749 

0115 

28.72 

'•759 

3453 

2  8. 70 

26 

.728 

4823 
6551 

28.80 

.738  X404 

28.75 

•749 

1838 

28.72 

•759 

5'75 

28.70 

27 

.728 

28.80 

•739 

0129 

28.75 

•749 

35'" 

28.72 

•759 

6897 

28.70 

28 

.728 

8279 

28.80 

•739 

1853 
3578 

28.75 

•749 

5284 

28.72 

•759 

8618 

28.69 

21) 

.729 

0006 

28.79 

•739 

28.75 

•749 

7007 

28.71 

.760 

0340 

28.69 

1    30 

1.729 

'734 

28.79 

'•739 

5303 

7028 

28.75 

1.749 

8730 

28.71 

1.760 

2062 

28.69 

31 

.729 

3461 

28.79 

•739 

28.75 

•75° 

0453 

28.71 

.760 

3783 

28.69 

32 

.729 

5189 

28.79 

•739 

8753 

28.75 

.750 

2176 

28.71 

.760 

5505 

2S.69 

1     33 

.729 

6916 

28.79 

.740 

0477 

28.75 

•75° 

3898 

28.71 

.760 

7227 

28.69 

34 

.729 

8644 

28.79 

•74° 

2202 

28.74 

•75° 

5621 

28.71 

.760 

8948 

28.69 

35 

1.730 

037' 

28.79 

1.740 

3927 

28.74 

1.750 

7344 
9067 

28.71 

1.761 

06 '/O 

28.69 

30 

.730 

2099 

28.79 

•74° 

5651 

28.74 

•750 

28.71 

.761 

=392 

28.69 

37 

.730 

3826 

28.79 

•74° 

7376 

28.74 

•75' 

0789 

28.71 

.76, 

4113 

28. 69 

38 

.730 

55S3 

28.79 

.740 

9101 

28.74 

•75' 

2512 

28.71 

.761 

5835 

28.69 

30 

.730 

7280 

28.79 

•74' 

0825 

28.74 

•75' 

4234 

28.71 

.761 

7556 

28.69     ; 

40 

1.730 

9007 

28.78 

'•74' 

2550 

28.74 

'•75' 

5957 

28.71 

1.761 

9278 

2S.69 

41 

•73« 

0735 

28.78 

•74' 

4274 

28.74 

•75' 

7680 

28.71 

.762 

0999 

2S.69 

42 

•73' 

2462 

28.78 

•74' 

5998 

28.74 

•75' 

9402 

28.71 

.762 

2721 

28.69 

43 

•73> 

4189 

28.78 

•74' 

7723 

2X.74 

.752 

1125 

28.71 

.762 

4442 

28.69 

44 

•731 

59'S 

28.78 

•74' 

9447 

28.74 

•752 

2847 

28.71 

.762 

6164 

28.69 

45 

1.731 

7642 

28.78 

1.742 

1171 

28.74 

1.752 

4570 

28.71 

1.762 

7885 

28. 69 

40 

•73' 

9369 

28.78 

•742 

2896 

28.74 

•752 

6292 

28.71 

.762 

9607 

28. 69 

47 

•732 

1096 

28.78 

•742 

4620 

28.74 

.752 

8015 

28.71 

.763 

1328 

28.69 

48 

•732 

2823 

28.78 

•742 

6344 

28.74 

.752 

9737 

28.71 

•763 

3050 

28.69 

40 

.732 

4549 

28.78 

.742 

8068 

28.74 

•753 

1460 

28.71 

.763 

477' 

28.69 

50 

1.732 

6276 

28.78 

1.742 

9792 

28.74 

'•753 

3182 

28.71 

'•763 

6493 

28.69 

51 

•732 

8002 

28.78 

•743 

1516 

28.73 

•753 

4904 

28.71 

.763 

8214 

28.69 

52 

•732 

9729 

28.77 

•743 

3240 
4964 

28.73 

•753 

6627 

28.71 

•763 

9936 

28.69 

53 

•733 

1455 

28.77 

•743 

28.73 

•753 

8349 

28.71 

.764 

1657 

28.69 

54 

•733 

3182 

28.77 

•743 

6688 

28.73 

•754 

0071 

28.70 

•764 

3379 

28.69 

55 

'■733 

4908 

28.77 

'•743 

8412 

28.73 

'•754 

1794 

28.70 

1.764 

5100 

28.69     1 

50 

•733 

663s 

28.77 

•744 

0136 
i860 

28.73 

■754 

3516 

28.70 

.764 

6821 

28.69     i 

57 

•733 

8361 

28.77 

•744 

28.73 

•754 

6960 

28.70 

•764 

8543 

28. 69 

58 

•734 

0087 

28.77 

•7^14 

3584 

28.73 

•754 

28.70 

.765 

0264 

28.69 

SO 

•734 

1813 

28.77 

•744 

5308 

28.73 

•754 

8682 

28.70 

•765 

1985 

28.69 

60 

'•734 

3539 

28.77 

1.744 

7031 

28.73 

'•755 

0405 

28.70 

1.765 

3707 

28.69 

582 


abolic  Orbit. 


TABLE  VI. 

For  rmdinigr  the  True  Aiioinsily  or  tlii'  Tiiiii'  from  tin-  Periliflion  in  a  Par.ibolic  Orhil. 


67^ 


kM. 


0405  I 

2127   ' 

5571 
7293 

9015 
0737 
2459 
41S1 
5903 

7625 

93+7 
1069 
2791 
45«3 
6235 

7957 
9679 
1401 
3>*3 

4844 
6566 
8288 
0010 
1731 


WIT.  1 ". 

28.70 
28.70 
28.70 
28.70 
28.70 

28.70 
28.70 
28.70 
28.70 
28.70 

28.70 
I  28.70 
'   28.70 

i   »>*-70 
;    28.70 

'    28.70 

:  28.70 

i  28.70 
28.70 
28.70 

28.70 

28.70 
28.70 
28.70 
28.70 


o 

o 

60 

60 


61 
61 
61 
61 
61 

61 


)9  3453 

59  5 '75 

59  ^**97 
9  8618 

0340 

2062 
3783 

5505 
7227 

60  8948 

0670 

-392  ; 
4113 

5835 
7556 

9278 
62  0999 
62  2721 
62  4442 
62  6164 

■62  7885 
■62  9607 
1328 
3050 

'63  477« 

763  6493 
763  8214 

763  9936 

764  1657 
764  3379 
764  5100 
764  6821 

764  8543 

765  0264 
765  1985 

765    3707 


28. 70 
28.70 
28.70 
28.69 
28.69 

28.69 
28.69 
28.69 
28.69 
28.69 

28.69 
28.69 
28.69 
28.69 
28.69 

28.69 
28.69 
28.69 
28.69 
i   28.69 

i  28.69 
28.69 
28.69 
28.69 
28.69 

28.69 
2S.69 
28.69 
28.69 
28.69 

i  28.69 
28.69 
28.69 
28.69 
28.69 

!  28.69 


V. 

68 

0 

69 

0 

70 

0 

7P         1 

1<.«  M. 

DIff.  1". 

IokM.   i 

DIff.  1". 

li>ii 

.M. 

DIff.  1". 

IokM. 

IHff.  1".  ' 

0 

i-7''5  3707 

28.69 

'•775  ''9^5 

28.69 

1.786 

028J. 
2006 

28.70 

-  - 

f.796  36^0 

28.73 

1 

.765  5428 

28.69 

•775  8706 

28.69 

.786 

28.70 

•796  5374 

28.73 

a 

•765  7150 

28.69 

.776  0427 

28.69 

.786 

3728 

28.70 

.796  7097 

28.73   , 

1   3 

.705  ****7> 

28.69 

.776  2149 

28.69 

.786 

5450 

28.70 

.796  8821 

28.73   , 

'   4 

.766  0592 

28.69 

.776  3870 

28.69 

.786 

717* 

28.70 

■-97  0545 

28.73   1 

5 

1.766  2314 

28.69 

1.776  ^591 

28.69 

I. -86 

8S94 

28.70 

1.797  2268 

28.73   1 

0 

.766  4035 

28.69 

•77''  73'3 

28.69 

.787 

0017 

28.70 

■797  3992 

28.73  : 

T 

.766  5756 

28.69 

.776  9"34 

28. 69 

■787 

2339 

28.70 

■797  57 '6 

28.73 

H 

.766  7478 

28.69 

•777  0755 

28.69 

.787 

4061 

28.70 

•797  7440 

28.73 

0 

.766  9199 

28.69 

•777  i477 

28.69 

.787 

5783 

28.70 

•797  9 '64 

28.73 

10 

1.767  0920 

28.69 

1.777  4198 

28,69 

1.787 

7506 

28.70 

1.798  0888 

28.73 

II 

.767  2642 

28.69 

•777  5920 

28.69 

■7!<7 

9218 

28.71 

.798  261 1 

28.73 

I'Z 

.767  4363 

28.69 

•777  7641 

28.69 

.788 

0950 

28.71 

•798  4335 
.798  6060 

28.73 

V.l 

.767  6084 

28.69 

•777  9363 

28.69 

.788 

2673 

28.71 

2X.73 

11 

.767  7805 

28.69 

.778  1084 

28.69 

.788 

4395 

28.71 

.798  7784 

28.73 

15 

1.767  95*7 

28.69 

1.778  2806 

28.69 

1.788 

6117 

28.71 

1.798  9508 

28.73  i 

10 

.768  1248 
.768  2969 

28.69 

•778  45*7 

28.69 

.788 

7840 

28.71 

•799  >J32 

28.74   ! 

17 

28.69 

.778  6248 

28.69 

.788 

9562 

28.71 

•799  2956 

2Sf^74   , 

IH 

.768  4691 

28.69 

■778  7970 

28.69 

■789 

1284 

28.71 

.799  4680 

28.74 

10 

.768  6412 

28.69 

.778  9691 

28.69 

.789 

3007 

28.71 

•799  6404 

28.74 

«() 

1.768  8133 

28.69 

1.779  i4«3 

28.69 

1.789 

4730 

28.71 

1.799  8128 

28.74 

21 

.768  9854 

28.69 

•779  3'40 

28.69 

.789 

6452 

28.71 

•799  9? 5 3 

28.74 

Tt 

.769  1576 

28.69 

•779  4862 

28.69 

.789 

8175 

28.71 

.800  1577 

28.74 

T.I 

.769  3297 

28.69 

•779  6578 

28.69 

.789 

9897 

28.71 

.800  3301 

28.74 

•u 

.769  5018 

28.69 

■779  8299 

28.69 

.790 

1620 

28.71 

.800  5026 

28.74   , 

25 

1.769  6740 

28.69 

1.780  0021 

28.69 

1.790 

3341 

28.71 

1.800  6750 

28.74   , 

20 

.769  8461 

28.69 

.780  1742 

18.69 

.790 

5065 

28.71 

.800  847; 

28.74 

27 

.770  0182 

28.69 

.780  3464 

28.69 

.790 

6788 

28.71 

.801  0199 

28.74  : 

28 

.770  1903 

28.69 

.780  5185 

28.69 

■79° 

8510 

28.71 

.801  1924 

28.74  I 

2U 

.770  3625 

28.69 

.780  6907 

28.69 

.791 

0233 

28.71 

.801  3648 

28.74 

:io 

1.770  S3a6 
.770  7067 

28.69 

1.780  8629 

28.69 

1.791 

1956 

28.71 

1.801  5373 

28.74  j 

31 

28.69 

.781  0350 

28.69 

.791 

3678 

28.71 

.801  7107 

28.74  1 

32 

.770  8788 

28.69 

.781  2072 

28.69 

.791 

5401 

28.71 

.801  8822 

28.74  : 

33 

.771  0510 

28.69 

•781  3793 

28.69 

.791 

7124 

28.71 

.802  0547 

28.75  ' 

34 

.771  2231 

28.69 

.781  5515 

28.69 

.791 

8847 

28.71 

.802  2271 

28.75 

35 

1.771  395* 

28.69 

1.781  7237 

28.69 

1.792 

0570 

28.71 

1.802  3996 

28.75 

30 

.771  5673 

28.69 

.781  8959 

28.69 

.792 

2293 

28.71 

.802  5721 

28.75  1 

37 

•77«  7395 

28.69 

.782  0680 

28.70 

.792 

4016 

28.72 

.802  7446 

28.75  , 

38 

.771  9116 

28.69 

.782  2402 

28.70 

.792 

5738 
7461 

28.72 

.802  9 17 1 

28.75   1 

39 

.772  0837 

28.69 

.782  4124 

28.70 

.792 

28.72 

.803  0896 

28.75  i 

40 

1.772  2559 

28.69 

1.782  5845 

28.70 

1.792 

9184 

28.72 

1.803  2^21 

28.75 

41 

.772  4280 

28.69 

.782  7567 

28.70 

•793 

0907 

28.72 

.803  4346 
.803  6071 

28.75 

42 

.772  6001 

28.69 

.782  9289 

28.70 

■793 

2630 

28.72 

28.75 

43 

.772  7722 

28.69 

.783  lOll 

28.70 

■793 

4354 

28.72 

.803  7796 

28.7s 

44 

•77a  9444 

28.69 

.783  2732 

28.70 

■793 

6077 

28.72 

.803  9521 

2^,75 

45 

1.773  "65 

28.69 

i^783  4454 

28.70 

1-793 

7800 

2^.72 

1.804  1246 

28.75 

40 

.773  i886 

28.69 

.783  6176 

28.70 

•793 

9523 
1246 

28.72 

.804  2971 

28.75 

47 

•773  4607 

28.69 

.783  7898 

28.70 

•794 

28.72 

.804  4697 

28.75 

48 

•773  6329 

28.69 

.783  9620 

28.70 

•794 

2969 

28.72 

.804  6422 

28.76 

49 

•773  8050 

28.69 

•784  134a 

28.70 

•794 

4693 

28.72 

.804  8147 

28.76 

50 

1.773  977" 

28.69 

1.784  3064 

28.70 

1.794 

6416 

28.72 

1.804  9873 

28.76 

51 

•774  »493 

28.69 

.784  4786 

28.70 

•794 

8139 
9862 

28.72 

.805  1598 

28.76 

52 

•774  32>4 

28.69 

.784  6508 

28.70 

•794 

28.72 

.805  3324 

28.76 

,  53 

•774  4935 

28.69 

.784  8230 

28.70 

•795 

1586 

28.72 

.805  5049 

28.76  : 

1  54 

■774  6657 

28.69 

•784  9952 

28.70 

•795 

3309 

28.72 

.805  6775 

28.76  ; 

1  55 

1.774  8378 

28.69 

1.785  1674 

28.70 

"•795 

mi 

28.72 

1.805  8500 

28.76  ; 

50 

•775  0099 

28.69 

•785  3396 

28.70 

•795 

28.72 

.806  0226 

28.76 

!  57 

•775  '8*« 

28.69 

.785  5118 

28.70 

•795 

8480 

28.72 

.806  1952 

28.76  i 

58 

•775  354* 
•775  5463 

28.69 

.785  6840 

28.70 

.796 

0203 

28.73 

.806  3677 

28.76  ' 

,  59 

28.69 

.785  8562 

28.70 

.796 

1927 

28.73 

.806  5403 

28.76  1 

1  00 

1.775  6985 

28.69 

1.786  0284 

28.70 

1.796 

3650 

28.73 

1.806  7129 

28.76 

am 


:-'^ 


TABLE  VI. 

For  finding  Iho  True  Anomaly  or  the  Tinu"  fnini  ilu'  IVriliolion  in  a  I'.iriiljolic  Orbit. 


V, 

0' 

72 

0 

73 

0 

74 

0 

IHff.  1". 

75 

0 

DilT  V 
28.95 

InK  M. 
1.806    7129 

Diir.  1". 
28.76 

l<>K  M. 
1.817    0765 

Dur.  1 ". 

1..K  M. 

2X.81 

1.827  a6o2 
.827  6315 
.827   8068 

28.XX 

1.X37   X6X6 

1 

.Hob  XXs? 

28.76 

.817    2494 

2X.X1 

2X.X8 

.X3X   0423 

2X.9, 

'i 

.Xo;  ojXi 

28.77 

.X17    4222 

2X.X2 

2X,S8 

,838    2160 

28.9^ 

3 

.X07  2307 

28.77 

.817    5951 

28.X2 

.827  9800 

28.88 

,838    3X98 

28.9^ 

4 

.807  4033 

28.77 

.817    7680 

2X.82 

.828   1533 

28.88 

.838    5635 

28,96 

5 

1,807  5759 

28.77 

I.817    9410 

28.82 

1.828   3266 

28.88 

1.838  737f 

28,96 

0 

.807  7485 

28.77 

.81X    1139 

2X.X2 

.X2X  4999 
,828  6732 

28,88 

.83X   9i:.y 

28,9(1 

7 

.807  9211 

28.77 

.818    286X 

2X.X2 

28, XX 

.X39  0847 

2X,9li 

8 

.XoX  0937 

28.77 

.XI 8  4597 

;X,X2 

.828   8465 

2X,X8 

.839  25X5 

28.9(1 

0 

.X08   2663 

28.77 

.XiX   6326 

28.x?. 

.829  01 98 

28.89 

.839  4323 

28.96 

10 

.XoX  4389 
.XoX  6116 

28.77 

1. 818  8056 

2X.S2 

1.X29   1931 
.X29  3(165 

28.89 

1.X39  6060 

28.96 

11 

28.77 

.818  97X5 

2X.X2 

28.89 

.X39  7798 

28.97 

i     >» 

.XoX   7X42 

28.77 

.XI9  1515 

2X.X3 

,X29  5-,9X 

28.89 

.X39  9536 

28.97 

1     13 

.808  9568 

28.77 

.XI9  3244 

2X.X3 

,X29  7131 

2X.89 

.Xp   1274 

28.97 

1     14 

.X09   1295 

28.77 

.8 1 9  4974 

2X.X3 

.X29  X865 

28.89 

.X40  3012 

28.97 

15 

.809   3021 

28.78 

1.8 1 9   6704 

Z8.83 

1.830  0599 

28.89 

1.840  4751 
.840  64X9 

28.97 

10 

.X09  4748 

28.78 

.819  8433 

28.83 

.830  2332 

28.89 

28.97 

17 

.X09  6474 

28.78 

.820  0163 

2X.S3 

.830  4006 

28,90 

.840  8227 

28,97 

18 

.X09  Xioi 

28.78 

.820   1X93 

28. S3 

.830  5800 

28.90 

.X40  9966 

2X.97 

10 

.809  9928 

28.78 

.820  3623 

28.83 

.830  7533 

28.90 

.841    1704 

28.98 

«0 

.810   1655 

28.78 

1.820  53S3 

28. 83 

1.830  9267 

28.90 

1.841    3443 

28,98 

Ul 

.Xio  33X1 

28.78 

.820  70X3 

28. 83 

831    ICOI 

28. 90 

.841    5182 

28,98 

U'Z 

.810   s'loX 

28.7X 

.X20  XX 1 3 

28. 84 

.831  273; 

2X,90 

.841   6921 

28,98 

23 

.810  6X35 

28. 78 

.821   0543 

28.X4 

.831  4470 

28,90 

.X41    8(159 

28,98 

•u 

.810  8562 

28.78 

.X21   2273 

28. S4 

.831  6204 

18,90 

.842  039X 

28,98 

25 

.811   02X9 

28.78 

1.X21   4003 

28.X4 

1.831  7938 

28.91 

1.842   2138 

28,98 

,    20 

.Xii   2016 

28.78 

.X21   5734 

28,84 

.831  9672 

2X.91 

.842   3X77 

28,99 

'    27 

.811    3743 

28.78 

.821   7464 

^li* 

.83a  1407 

28.91 

.842  5616 

28,99 

28 

.8-1   S470 

28.79 

.821   9194 

28. X4 

.832  3141 

28.91 

.842  7355 

2X.99 

20 

.1.   I   7197 

28.79 

.822  0925 

28.84 

.83-  •X76 

28.91 

.842  9095 

28,99 

30 

.Xii   8924 

28.79 

1.822  2656 

28.X4 

1.832  66ii 

28.91 

1.843  0834 

28.99 

31 

.812  0652 

28.79 

.X22  43X6 
.822  6117 

Hi* 

.832  8345 

28.91 

•843   i574 

28,99 

»i 

.812  2379 

2X.79 

28.85 

.833   0080 

28.92 

•843  43>3 
.843  6053 

29,00 

33 

.812  4106 

28.79 

.822  784X 

28.85 

.833    ,X.5 

28,92 

29,00 

34 

.812  5834 

28.79 

.822  9578 

28.85 

•833   3550 

28,92 

•843  7793 

29,00 

35 

.812  7561 

28.79 

1.823   '3°9 

28.85 

1.833   52X5 

28,92 

■•843  9533 

29.00 

30 

.812  9289 

28.79 

.823   3040 

28.85 

.833  7020 

28.92 

.844  1273 

29.00 

37 

.813   1016 

28.79 

.823  4771 

28. 85 

•833  «755 

28,92 

.844  3013 

29.00 

1     38 

.813   2744 

28.79 

.823   6502 

28.85 

.834  0491 

28,92 

•844  4753 

29.00 

1    30 

.813  4472 

28.79 

.823   8233 

28.85 

.834  2226 

28,92 

.844  6494 

29.01 

40 

.813  6199 

28.80 

1.823  9965 

28.85 

1.834  3961 

2X.92 

1.X44  8234 

29.01 

41 

.813  7927 

28.80 

.824  1696 

28.85 

.834  5697 

28.93 

.844  9974 

29.01 

42 

.813  9655 

28.80 

.824   3427 

28.86 

■834  743» 

28,93 

•845   >7I5 

29.01 

43 

.8,4  1383 

28.80 

.824  S159 

28.86 

.834  9168 

28.93 

•845   3456 

29.01 

44 

.814  3111 

28.80 

.824  6890 

28.86 

.835  0904 

28.93 

.845   5196 

29.01 

45 

.814  4.839 
.814  6567 

28.80 

1.824  8622 

28.86 

1.835   2640 

28.93 

1.845  6937 

29.01 

40 

28.80 

.825  0353 

2X.86 

.835  4376 

28.93 

.845   X678 

29.02 

47 

.814  8295 

28.80 

.825   20X5 

28.86 

.835  6112 

28,93 

.846  0419 

29.02 

48 

.815  0023 

28.80 

.825   3816 

28.86 

.835   7848 

28,93 

.846  2160 

29,02 

40 

.815   1751 

28.80 

.825   5548 

28.86 

.835  9584 

28,94 

.846  3901 

29,02 

50 

.815  3479 
.815   5208 

28.80 

1.825  7280 

28.86 

1.836   1320 

28.94 

1,846  5643 

29.02 

51 

28.81 

.825  9012 

28.87 

.836   3056 

28.94 

.846  7384 

29.02     ' 

52 

.81C  6936 

28.81 

.826  0744 

28.87 

.836  4792 

28.94 

.846  9125 

29.03 

53 

.815   8664 

28.81 

.826  2476 

28.87 

.836  6529 

28.94 

.847  0867 

29.03     , 

54 

.816  0393 

28.81 

.826  4208 

28.87 

.836  8265 

28.94 

.847  2609 

29.03 

55    1 

.816    2I2I 

28.81 

1.826  5940 

28.87 

1.837  0002 

28.94 

1.847  4350 

29.03 

50 

.816    3850 

28.81 

.826  7673 

28.87 

.837   1739 

28.95 

.847  6092 

29.03 

57 

.816    5578 

28.81 

.826  9405 

28.87 

•837  3475 

28.95 

.847  7834 

29.03 

58 

.816    7307 

28.81 

.827   1137 

28.87 

.837  5212 

28,95 

.847  9576 

29.03 

50 

.816    9036 

28.81 

.827  2870 

28.87 

.837  6949 

28.95 

.848   1318 

29.04 

00  |i 

.817    0765 

28.81 

1.827  4602 

28.88 

1.837  8686 

28.9s 

1.848   3060 

29.04 

684 


iiholic  Orbit. 


TABLE  VI. 

For  ffniliiiK  tlic  Tnic  Anoiimly  or  lla-  Tiinc  from  tin-  Pcrilielion  in  a  Purulxilio  Orbit, 


V, 
0' 

76 

D 

77° 

78 

0 

79 

0 

log  M. 

wir.  1". 

19.04 

l»K 

M. 

7769 

mrr.  i". 
29.14 

i.>g 

M. 

Wff.  1". 
29.15 

l->K 

.M. 

iMir.  y. 
2937 

1.X4S 

3060 

1.858 

1.869 

?,8  57 

'•879 

8369 

1 

.X4S  4803 

29.04 

.85,X 

95'7 

29.14 

.869 

a6|2 

6367 

29.25 

.880 

0131 

2937  , 

3 

.X4X 

6545 

29.04 

.859 

1266 

29.14 

.869 

29.25 

.880 

1894 

19.38 

» 

.X4S 

8287 

29.04 

■  8^9 

3014 

29.14 

.869 

8122 

29.25 

.880 

36  5f' 

19.38 

4 

.849 

0030 

29.04 

.859 

4763 

29.15 

.869 

9878 

29.16 

.880 

54' 9 

29.38  1 

5 

1.84., 

'773 

29.04 

1.859 

6512 

29.15 

1.870 

1613 

19.26 

i.88o 

7182 

19.38  ' 

0 

.84.; 

35'5 

29.05 

.859 

8260 

29. 1 5 

.870 

i^h 

29.26 

,880 

8945 

19. 38 

T 

.849 

5258 

29.05 

.X60 

0009 

29.15 

.870 

5'44 

19.16 

.881 

0708 

19.39 

8 

.849 

7001 

19.05 

.86d 

1758 

29.15 

.870 

6900 

19.16 

.881 

2471 

19.39 

0 

.849  8744 

29.05 

•.860 

3507 

29.15 

.870 

8656 

19.26 

.881 

4235 

29. 39 

10 

1.8^0 

0487 

29.05  . 

1.860 

5256 

29.15 

1.871 

0412 

19,17 

1.881 

5998 

29.39 

11 

.8^0 

2231 

29.05 

.860 

700S 

29.16 

.X71 

2168 

19.27 

.881 

7762 

19.39 

Vi 

.S50 

3974 

29.06 

.860 

8755 

29.16 

.871 

3914 

19.17 

,88 1 

9516 

19.40 

13 

.850 

S7'7 

29.06 

.861 

0505 

29.16 

.871 

5O81 

19.17 

.882 

1290 

19.40  ; 

14 

.X50 

7461 

19.06 

.8bi 

2254 

29.16 

.87. 

7437 

19.28 

.882 

3°54 

29.40   : 

15 

1.8^0 

9204 

29.06 

1.861 

4r.04 

29. 16 

1.871 

9 '94 

29.18 

1.882 

4818 

29.40  ; 

lU 

.85, 

0948 

29.06 

.8f  I 

5754 

29.16 

.871 

0950 

19.1  s 

.882 

6581 

19.41 

n 

.85, 

2692 

29.06 

.861 

7504 

29.17 

.872 

1707 

19.18 

.882 

8347 

19.41 

18 

.85, 

4436 
6i8o 

29.07 

.861 

9»54 

29.17 

•2'" 

4464 

19.18 

.883 

01 11 

19.41 

10 

.85, 

29.07 

.862 

1004 

29.17 

.872 

6221 

29.29 

.883 

1876 

29.41 

20 

1.8^1 

79*4 
9668 

29.07 

1.862 

»754 

29.17 

1.872 

7979 

29,29 

1.883 

3641 

29.42 

•il 

.851 

29.07 

.862 

45°5 

29.17 

.872 

9736 

29.29 

.S83 

5406 

29.41 

'Z'Z 

.85a 

1412 

29.07 

.862 

6^55 

29.18 

.873 

'493 

29.29 

.883 

7171 

19.41 

33 

.852 

3157 

29.07 

.862 

8006 

29.18 

.873 

325' 

29.29 

.883 

8937 

29-42 

34 

.85Z 

4901 

29.07 

.862 

9756 

29.18 

.873 

5008 

19.30 

.884 

0702 

19.42  1 

35 

1.852 

6646 

29.08 

1.863 

1507 

29,18 

1.873 

6766 

29.30 

1.884 

2468 

29-43  1 

3» 

.8,2 

8391 

29.08 

.863 

3258 

29.18 

.873 

8524 

29.30 

.884 

4233 

2943 

37 

.8,3 

0135 

29.08 

■l^J 

5009 

29,18 

.874 

0282 

29.30 

.884 

5999 

2943 

38 

.853 

1880 

29.08 

■It^ 

6760 

29.19 

.874 

2041 

29.30 

.884 

7765 

19.43 

30 

.853 

3625 

29.08 

.863 

8512 

29.19 

•874 

3799 

29.31 

.884 

953' 

19.44 

30 

••853 

5370 

29.09 

1.864 

0263 

29.19 

..874 

5557 

29.31 

1.885 

1297 

29.44 

31 

.853 

711S 

29.09 

.864 

2015 

29.19 

.874 

7316 

29.31 

.X85 

3064 

29.44  ' 

:i3 

.853 

8861 

29.09 

.864  3766 

29.19 

•874 

9074 

29.31 

.885 

4830 

29-44 

33 

.854 

0606 

29.09 

.864 

5518 

29.20 

.875 

o8t3 

29.31 

.885 

''597 

2945 

31 

.854 

2351 

29.09 

.864 

7270 

29.20 

.875 

2592 

29.32 

.885 

8364 

29.45 

35 

..854 

4097 

29.09 

1.864 

9022 

29,20 

1-875 

4351 
6111 

29.32 

1.886 

0131 

29.45 

30 

.854 

5843 

29.10 

.865 

0774 
2526 

29.20 

-875 

29.32 

.886 

1898 

29.45 

3T 

.854 

7588 

29.10 

•^^5 

29.20 

■875 

7870 

29.32 

.886 

3605 

29.45 

38 

.854 

9334 

29.10 

.865 

4278 

29,20 

.875 

9629 

29.32 

.886 

5432 

7 

30 

.855 

1080 

29.10 

.865 

6030 

29.21 

.876 

1389 

29-33 

.886 

7200 

29t6 

40 

1.855 

2826 

29.10 

1.865 

778; 
953^ 

29.21 

1.876 

3148 

29.33 

1.886 

8967 

29.46 

41 

.855 

4572 

29.10 

.865 

29.21 

.876 

4908 

19.33 

.887 

0735 

19.46  i 

43 

.855 

6319 

29.11 

.866 

1288 

29.21 

.876 

6668 

29-33 

.887 

2503 

29-47  1 

43 

.855 

8065 

29.11 

.866 

3041 

29.21 

.876  8428 

29.33 

.887 

4271 

29-47  1 

44 

.855 

9811 

29.11 

.866 

4794 

29.22 

.877 

0188 

29-34 

.887 

6039 

29-47  i 

45 

1.856 

1558 

29.11 

1.866 

6547 

29.22 

1.877 

'949 

29-34 

1.887 

7807 

29.47  ' 

40 

•^56 

3305 

29.11 

.866 

8301 

29.22 

.877 

3709 

29,34 

.887 

9576 

19.48 

47 

.856 

505* 

29.11 

.867 

0054 

29.22 

.877 

5470 

29.34 

.888 

'344 

19.48 

48 

.856 

6799 

29.12 

.867 

1807 

29.22 

•877 

7230 

29.34 

.888 

3113 

19.48 

40 

.856 

85+6 

29.12 

.867 

3561 

29.23 

.877 

8991 

29.35 

.888 

4882 

29.48 

50 

1.857 

0293 

29,12 

1.867 

53J4 

29-23 

1.878 

0752 

29-35 

1.888 

6651 

29.48 

51 

.857 

2040 

29.12 

.867 

7068 

29.23 

.878 

2513 

29-35 

.888 

8420 

29.49 

53 

.857 

3787 

29.12 

.867 

8822 

29.23 

.878 

4275 

29-35 

.889 

0189 

29-49 

53 

.857 

5534 

29.12 

.868 

0576 

29.23 

.87S 

6036 

29-35 

■It^ 

'959 

29.49 

54 

.857 

7282 

29.13 

.868 

2330 

29.24 

.878 

7797 

29.36 

.889 

3728 

29.49 

;  55 

1.857 

9030 

29.13 

1.868 

4084 

29.24 

1.878 

9559 

29.36 

1.889 

5498 

29.49 

50 

.858 

0777 

29.13 

.868 

5839 

29.24 

.879 

1321 

29.36 

.889 

7168 

29.50 

57 

.858 

2525 

29.13 

.868 

7593 
9348 

29.24 

.879  3082 

2936 

.889 

8038 

29.50 

58 

.858 

4173 

602l 

29.13 

.868 

29.24 

.879 

4844 

29.36 

.890 

0808 

19.50 

50 

.858 

29.13 

.869 

1102 

29.25 

.879 

6606 

29-37 

.890 

2578 

29.51 

GO 

1.858 

7769 

29.14 

1,869  2857 

29.25 

1.879  8369 

29,37 

1.890 

4349 

29.51 

585 


TABLE  VI. 

For  liiiilinK  llii^  True  Aiioimilv  ur  tin-  Tiiiu'  I'ntni  tlio  IVrilii'lioi)  in  a  I'liraliolic  Orliit. 


€ 


I'. 

i.Xi^o 

80 

° 

I..V 

81 

[M. 

Dinr.  1". 

29.66 

82 

fj 

83 

!..»;  .M. 
1.922    5548 

0 

M. 

4U'> 

IHIT.  1". 
^9-51 

Ion 
1.9  I  I 

M. 

7893 

iMir.  1". 

29. Xi 

wrr.  1". 

o 

1.901 

0X41 

19.99 

1 

.X.>o 

M19 

29  SI 

,901 

2(121 

29.66 

.911 

96X2 

19.x  1 

■92*    7347 

19.99 

u 

.S.;o 

7X90 

29.51 

.901 

4400 

29  66 

.911 

1471 

19. Xi 

.912    9147 

30. OJ 

:i 

.Syo 

9661 

19-51 

.901 

61X0 

29  66 

.911 

3261 

29-83 

-923    0947 

30.00 

4 

.S91 

I43» 

29.  SI 

.901 

7960 

19.67 

.911 

5050 

29.83 

.923    1747 

30.00 

a 

l.Xi;l 

-5103 

»9-5» 

1.90 1 

9740 

19.67 

1.9 1  2 

6X40 

^.9.83 

1-923    4548 

30.01 

n 

.Xyi 

4V74 

19.52 

.9^2 

1521 

29.67 

.912 

X630 

2984 

.923    6348 

30.01 

7 

.X9I 

^745 

29.51 

.901 

3301 

29.67 

.913 

0420 

29.  X4 

-92?    8149 

30.01 

1        N 

.XVI 

8^17 

»953 

.901 

50X2 

19.68 

.913 

2211 

29-84 

-913    9950 

30.01 

1       0 

.X.;z 

0189 

»9-53 

.901 

()862 

19.68 

.913 

4001 

2984 

.924    1751 

30,02 

10 

I.S92 

1061 

»9-53 

1.901 

8643 

19.68 

1.913 

579* 

29-85 

1.924    3552 

30.01 

!    li 

.Xyi 

3«^3 

*9-53 

.903 

0424 

29.69 

■913 

7583 

29-85 

-924    5354 

30.03 

1'^ 

•  Syl 

t,(<o<i 

19.54 

.903 

2105 

29.69 

.913 

9374 

29-85 

.924    7155 

30.03 

i:i 

.S92 

mi 

»9-54 

.903 

39X7 

29.69 

.914 

1165 

29-85 

-924    8957 

30.03 

It 

.Xyi 

9149 

*954 

.903 

5768 

29.69 

.914 

2956 

29. X6 

.925    0759 

30.03 

irt 

lJ,)H 

0921 

19.54 

1.903 

7550 

19.70 

1.914 

4748 

29.86 

1.925    2561 

30.04 

i    '" 

.X,,5 

1695 

19- 5  5 

•9-33 

933* 

19.70 

.914 

6540 

29.  X6 

-925  43''4 

30.04 

!     17 

.«.>■! 

44''7 

*9-55 

.904 

"4 

1896 

29.70 

.914 

833> 

29.X7 

.925   6166 

3004 

IH 

.X.;3 

6240 

1955 

.904 

29.70 

.915 

012A 
1916 

29-87 

•  925   7969 

30.05 

10 

.X93 

8013 

»955 

.904 

4678 

19.71 

•9'S 

19.87 

.925  9771 

30.05 

•zo 

i.X.n 

9787 

19.56 

1.904 

6461 

19.71 

1-915 

3708 

29-87 

1.916   1575 

30,05 

1  *-*' 

.X,H 

1560 

29.56 

.904 

X243 

29.71 

.915 

5  5o< 

19.88 

.926    337X 

30.06 

a« 

•X'H 

^3U 

29,56 

.905 

0026 

29.71 

-915 

7294 

29.S8 

.926   51X1 

30.06 

1     «3 

.X94 

S  loX 

29.56 

.905 

1X09 

29.72 

.915 

90X7 

29.X8 

.926  69X6 

30.06 

'H 

.Xy4 

6X82 

19-57 

.905 

359* 

29.72 

.916 

0880 

19.89 

.926   87X9 

30.07 

j     '^5 

I.X.H 

86^6 

29-57 

1.905 

5376 

29.72 

1.9 1 6 

2673 

19.89 

1.927  0591 
.927   2398 

30.07 

1     2U 

•  X'^i 

04^0 

29-57 

.905 

7 '59 

29-73 

.9 1 6 

4466 

19.89 

30.07 

i     «7 

.X.,s 

2204 

»9-57 

.905 

8943 

29-7  3 

.916 

6i6o 

29.90 

.927  J.202 
.927  6007 

30.08 

1     «H 

.X9<; 

3979 

29.58 

.90(1 

0726 

29-73 

.916 

^t^t 

29.90 

30.08 

1   at) 

.X95 

5753 

29.58 

.906 

2510 

29-73 

.916 

9848 

29.90 

.927   7X11 

30.08 

:i(> 

1.X95 

7,-28 

19.58 

1.906 

4294 

29.74 

1.917 

1642 

29.90 

1.927  9616 

3r.rS 

:ii 

.X9S 

9303 

29.58 

.906 

6o'9 

2974 

.917 

3436 

29.91 

.928    1422 

3-9 

1    :w 

.X96 

1078 

19.59 

.906  7X63 

29.74 

.917 

523' 

29-91 

-928   3227 

30.09 

»a 

.X96 

2854 

19.59 

.906 

964X 

29-74 

.917 

7025 

29.91 

.92S    5032 

30.09 

34 

.X96  4628 

19.59 

.907 

.432 

*9-7S 

.917 

8820 

19.91 

.918  6838 

30,10 

35 

1.X96  6404 

29.59 

1.907 

3217 

29-75 

1. 918 

0615 

29.92 

1.928   8644 

30. JO 

3U 

.896 

8180 

29.60 

.907 

5002 

29-75 

.918 

2410 

29.92 

-929  0450 

30.10 

37 

.X9O 

995  5 

29.60 

.907 

6787 

29-75 

.918 

A206 
6001 

29.92 

.929  2256 

3  c.  1 1 

38 

.X97 

1732 

29.60 

.907 

«573 

29.76 

.918 

29-93 

.929  4063 

-,c.ll 

30 

.X97 

3508 

29.60 

.908 

0358 

29.76 

.918 

7797 

2993 

•929  5869 

30.11 

4U 

1.X97 

5284 

29.61 

1.908 

2144 

29.76 

1.918 

9593 

2993 

1.929  7676 

3c.ll 

41 

.X97 

7060 

29.61 

.90X 

393" 

29-77 

.919 

1389 

29.94 

.929  9483 

50.12 

4'Z 

.X97 

8837 

19.61 

.908 

5716 

29-77 

.919 

31X5 

29-94 

.930   1 291 

50.12 

43 

.X9X 

0614 

29.61 

.908 

7502 

29-77 

.919 

4982 

29.94 

.930  309X 

-,c.i3 

44 

.898 

2390 

29.62 

.908 

9288 

29-77 

.919 

6778 

29.94 

.930  4906 

30.13 

45 

1.X98 

4168 

29.62 

1.909 

.075 

29.78 

1. 919 

8575 

29.95 

1.930  6713 

3c,  13 

40 

.89X 

5945 

19.62 

.909 

2X62 

29.78 

.920 

0372 

29-95 

.930  8521 

3C.13 

47 

.89X 

7722 

29.62 

.909 

4648 

29.78 

.920 

2169 

29.95 

-93'   0330 

30  14 

48 

.X98 

9500 

29.63 

.909 

6436 

29.78 

.920 

3966 

29.96 

.931    213X 

30.14 

40 

.899 

1277 

29.63 

.909 

8123 

25.79 

.910 

5764 

29.96 

-93 «    3946 

30,14 

50 

1.899 

3055 

19.63 

1. 910 

0010 

29.79 

1.920 

7561 

19.96 

'-93'    5755 

30.15 

51 

.899 

66 1 1 

29.63 

.910 

1798 

29.79 

.920 

9359 

29.97 

-93'    7564 

3c.  15 

52 

.X99 

29.64 

.910 

35!<5 

29.80 

.921 

"57 

29.97 

•93'   9373 

30.15 

53 

.899 

8389 

29.64 

.910 

5373 

29.80 

-921 

2956 

29.97 

.932    1183 

30.16     ■ 

54 

.900 

0168 

29.64 

.910 

7161 

29.80 

.921 

4754 

29.98 

-932  2992 

30.16 

.   55 

1.900 

1946 

29.64 

1. 910 

8949 

29.80 

1. 911 

6552 

29.98 

1.932  4802 

30.16 

50 

.900 

3715 

29.65 

.911 

0738 

29.81 

.911 

835" 

29.98 

.932  6612 

30,17 

57 

.900 

5504 

29.65 

.911 

2526 

29.81 

.922 

0150 

29.98 

.932   8422 

3C.17 

58 

.900 

7283 

29.65 

.911 

43  >  5 

29.81 

.921 

1949 

29-99 

-933  0232 

30.17 

50 

.900 

906a 

29.66 

.911 

6104 

19.82 

.911 

3748 

19.99 

•933   2043 

30.18 

60 

1. 90 1 

0841 

29.66 

1.911 

7893 

19.82 

1.911 

5548 

19.99 

1-933    3853 

30.18 

580 


nilxtlic  Orltit. 


TABLE  VI. 

Kiir  liiiiliii),'  till'  TriK'  Atiom^ily  nr  llir  'I'iiiii'  iVkiii  tlu'  Pcrilirlion  in  a  Par.ilMilii'  ()rlii(. 


83 

) 

•kM.         j 

iMir.  r 

I  sux 

29.9., 

»  7U7   1 

29.99 

1  <>i47   ' 

30.0  J 

1  0';47 

30.03 

i   *747   ' 

30.00 

?   454i<   ' 

30.01 

\  f'U» 

30. '11 

X  K149  1 

30.',  I 

1  9750  ' 

30.02 

4   '751    1 

30,01 

4   l^-?* 

30.02 

4    ^1H 

30.03 

4   7i>S 

30-C'l 

4  !<957 

30.03 

5   °759 

30.03 

5   »5''l 

30.04 

i;  4lf'4 
5   6166 

30.04 

3004 

5   7969 

30.05 

5   977* 

30.05 

6   M75 

6    3V» 

30,05 

30. c6 

6    5181 

30.06 

6   6.)86 

30.06 

6   8789 

30.07 

7  OS93 
7    1V>» 

30.07 

3C,07 

7  4101 
7  6007 

3o.o.>f 

30.0X 

7  7«n 

30.08 

7  9616 

30.08 

8    1422 

30.C9 

8   3227 

30,09 

8    ^032 

3o.o<) 

8   6838 

30.10 

8    8644 

30. iO 

9  0450 

30.10 

9  2256 

3c.  1 1 

9  4061 

3c.  11 

9  S«69 

30.H 

9   7676 

30.12 

9  94*' 3 

30.12 

D     I  29 1 

30.1a 

0  3098 

30.13 

0  4906 

30.13 

0  6713 

30.13 

0  8521 

30.13 

I  0330 

30  14 

I  2138 

30.14 

I  3946 

30.14 

I    1755 

30.15 

«    7564 

3C.I5 

«   9373 

30.15 

2    1183 

30.16 

2  2992 

30.16 

2  4802 

30.16 

2  6612 

30.17 

2   8422 

3C.I7    ' 

3  0232 

3"'i 

3   2043 

j   30.18 

3853  i  30.18 


84 

0 

85 

86 

0 

87 

r, 

1'..:  M. 

•933   3><53 

IMfl.  1". 

30.18 

l"K 

M. 

nitr.  f 
30.38 

I..K  M. 

f.955  1602 

iiiir  1". 
30.59 

l"K 

M. 

Kill   1". 
30.U2 

'  '>44 

2856 

1.966 

3140 

1 

•933    S<'"4 

30.18 

•944 

4678 

30.38 

•955  ■f43'' 

30.60 

.966 

4990 

30.82 

u 

•933   747; 

30.19 

•94  4 

<'50» 

30  39 

•955   •'2T4 

30.60 

.966 

6839 

30.83 

:i 

933  91X7 

30.19 

•'U4 

8}2< 

0148 

30.39 

.955   8110 

30.60 

.966 

8689 

30.83 

1 

9  34   '09'* 

30.19 

•945 

30.39 

95  5  9946 

30.61 

,967 

0539 

30.84      ^ 

5 

1.934  19  >o 

30.20 

"•945 

1972 

30.40 

1.956   1783 

30.61 

f.967 

2,89 

3084 

n 

...34  4711 

30.20 

•945 

3796 

30.40 

.956    3619 

30.61 

.967 

424-3 

30.84 

7 

934   <'^33 

30.20 

•945 

5620 

30.40 

.956   5456 

3C.62 

.967 

6090 

30.85 

N 

9  34   X34<' 

3021 

•945 

-444 

♦130.41 

.956  7294 

30.62 

.967 

794 « 

30.85 

U 

.935   0158 

30.21 

•945 

9269 

30.41 

.956  9131 

30.63 

.967 

9792 

30.85 

10 

1.935    1971 

30.21 

■  ••)46 

1094 

30.41 

"•957   '^9'^9 

30.63 

i.9<iS 

.644 
34')" 

30.86 

II 

•935    37X4 

30.22 

.946 

2919 

30.42 

•957    2807 

3o.(>3 

.9h8 

30.86 

n 

•935    5597 

30.22 

.746 

4744 

30.42 

•957  4<>45 

30.64 

.968 

5  347 

30.87 

1:1 

•93  5   74 "o 

30.22 

.946 

<'5''9 

30.42 

•957  64X3 

30.64 

.968 

7200 

30,87 

II 

•93  5   9113 

30.22 

.946 

I*  395 

30.43 

957  83'H 

30.64 

.968 

9052 

30.87 

i.'i 

1.936    1037 

30.23 

'947 

0221 

30-43 

1.958  0160 

30.65 

1.969 

0905 

30.88 

lit 

.93(1   2851 

30.23 

•'n7 

2047 

3044 

.958   1999 

30.65 

.9(19 

2757 

30.88       '■ 

17 

.936  4665 

30.23 

•947 

3X73 

3044 

.958    3839 

30.66 

■'>'P 

4610 

30.89 

IH 

.936  6479 

3o^i4 

•947 

5699 

30.44 

.958   5678 

30.66 

•9''9 

30.89 

l» 

.936  8293 

30.24 

947 

7ji6 

30-45 

.958  7518 

30.66 

.969 

8317 

30.89 

'^0 

1.937  0108 

30.24 

'  947 

9353 

30.45 

1.958  9358 

30.67 

1.9-0 

0171 

30.90 

•il 

•9  37    192=1 

30.25 

.948 

iiSd 

30.45 

.95<)    1198 

30.67 

.970 

2025 

30.90 

•ii 

•937   3737 

30.25 

■'n^ 

3007 

30.46 

•959   3038 

50.67 

.970 

3879 

30.91 

'i'.i 

•937    5551 
•9  37  73"i< 

30.25 

.948 

4«34 

30.46 

•959  4'*79 

30.68 

-V 

5734 
75S9 

30.91 

•a 

30.26 

.948 

6602 

30.46 

•959  <'720 

30.68 

.970 

30.91 

•i't 

1.937  91S4 

JO.  26 

1.948 

8490 

30.47 

1.959   8561 

30.69 

1.970 

9443 

30.92       ; 

'U\ 

.938  0999 

30.26 

•94'> 

0318 

30.47 

.960  0402 

30.69 

•97' 

1  299 

30^92      I 

'i7 

.938   2815 

3-    i7 

•949 

2146 

30-47 

.960   2243 

30.69 

•97' 

3'54 

30.93      i 

!iH 

.938  4632 

v^.27 

•949 

397  5 

30.48 

.960  40S5 

30.70 

.971 

5010 

3093 

•i\i 

.938   6448 

30.27 

•949 

5804 

30.48 

.960  5927 

30.70 

•97' 

6866 

30.93 

.10 

1.938   8264 

30.28 

1.949 

7633 
9462 

30.48 

1.960  7769 

30.70 

1.971 

8t22 

3094 

31 

.939  0081 

30.28 

•949 

30.49 

.960  9612 

30.71 

•972 

0578 

30.94 

M 

.939   1898 

30.28 

.950 

1291 

30.49 

.961    1454 

30.71 

•972 

2435 

30<)5 

;i:i 

•939   37 'S 

3C.29 

•950 

3»n 

30.50 

.961    3297 

30.71 

.972 

4292 

30.95 

34 

•939   5533 

30.29 

•950 

495" 

30.50 

.961    5140 

30.72 

•972 

6149 

30^95      j 

35 

1.939  7350 

30.29 

1.950 

6781 

30.50 

1.961    6983 

30.72 

1.972 

8006 

30.96      j 

3» 

.939  9168 

30.30 

.950 

8611 

30.51 

.961    8827 

30^73 

.972 

9864 

30,96 

37 

.940  0986 

30.30 

•951 

0441 

30.51 

.962  0671 

3073 

•97  3 

1  -ti  ■* 

30.97 

3H 

.940  2804 

30.30 

■  951 

2272 

30.51 

.962   2515 

30.73 

•97  3 

3580 

30.97 

3U 

.940  4623 

30.31 

•95  • 

4103 

30.52 

.962  4359 

30^74 

•97  3 

5438 

30.97 

10 

1.940  6441 

30.31 

1. 951 

lUt 

30.52 

1.962  6203 

30^74 

'•97  3 

7297 

30.98 

11 

.940  8260 

3031 

•951 

30.52 

.96.   8048 

30^75 

•97  3 

9156 

30.98 

Vi 

.941   0079 

30.32 

•95' 

9597 

30^  5  3 

.962   9S93 

30^75 

•974 

1015 

30.99 

13 

.941    1898 

30.32 

.952 

1429 

30-53 

.963    1738 

30^75 

•974 

2X74 

30.99      1 

II 

•94«    37"  7 

30.32 

.952 

3261 

30-53 

.963   3583 

30.76 

•974 

4734 

30.99       ; 

15 

'•94'   5537 

■3o^33 

1.952 

5093 

30.54 

1.963    5429 

30.76 

'•974 

6593 

31.00       : 

1(1 

•94"   7357 

3033 

•952 

69    ■ 

30-54 

.963   7275 

30^77 

•974 

8454 

31.00 

17 

.941   9177 

30-34 

.952 

8758 

30^55 

.963  9121 

30^77 

•975 

0314 

3;. 01 

IH 

.942  0997 

3034 

-95  3 

0591 

30.55 

.964  0967 

3077 

-975 

2174 

31.01 

lU 

.942  2817 

3o^34 

•953 

2424 

30-55 

.964  2814 

30.78 

-975 

4035 

31.01 

50 

1.942  4638 

30.35 

'•953 

4257 

30.56 

1.964  4660 

30.78 

'•975 

5896 

31.02 

51 

.942  6459 

30.35 

•953 

6091 

30.56 

.964  6507 

30.78 

•975 

7757 

31.02 

5'i 

.942  8280 

30-35 

•95  3 

7924 

30.56 

.964  8354 

30.79 

•975 

9619 

31.03 

53 

.943  01 01 

30.36 

•95  3 

9758 

30.57 

.965   0202 

30.79 

.976 

1481 

31.03 

54 

•943   '913 

30.36 

■954 

1592 

30^57 

.965  2050 

30.80 

.976 

3343 

31.04 

55 

«-943   3744 
•943   5566 

30.36 

"•954 

3427 

3057 

1.965   3897 

30.80 

1.976 

5205 

31.04     . 

50 

30^37 

•954 

5262 

30.58 

.965    5746 

30.80 

.976  7067 

31.04 

57 

•943   7388 

3037 

•954 

7096 

30.58 

•965   7594 

30.8, 

,976 

8930 

31.05 

58 

•943  9m 

30-37 

•954 

0766 

30-59 

.965  9442 

30.81 

-977 

0793 

31.05 

59 

•944  >033 

30.38 

•955 

30.59 

.966   1291 

30.81 

-977 

2656 

31.06 

00 

1.944  1856 

30.38 

'•955 

2602 

30.59 

1.966   3140 

30.82 

«-977 

4520 

31.06 

587 


f5 


/■«?>;■  V 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Oibit. 


588 


V. 
0' 

88 

0 

S9 

0 

90 

0 

91 

0 

logM. 

Diff.  1". 

log  M. 

1.988    6789 

Diff.  1". 

log  M. 

Diff.  1". 

logM. 

Diff.  1". 

1.977  4520 

31.06 

31.31 

2.000 

0000 

31.50 

2.01 1 

4203 

1 

■977  6383 

31.06 

.988  8668 

31.32 

.000 

1895 

3'-59 

.011 

61 15 

3'-87 

2 

•977  8247 

31.07 

.989  0548 

31.32 

.000 

3790 

3'-59 

.Oil 

8027 

3i.S,S 

3 

.978  0112 

31.07 

.989  2427 

3'-33 

.000 

5686 

31.60 

.01 1 

9940 

31.88     ' 

4 

.978   1976 

31.08 

.989  4307 

3«-33 

.000 

7582 

31.60 

.012 

1853 

31.89     1 

3 

1.978  3841 

31.08 

1.989  6187 

3>-34 

2.000 

9478 

31.60 

2.012 

3766 

31.89 

6 

.978   5706 

31.08 

.9X9  8067 

31-34 

.001 

>375 

31.61 

.012 

5680 

31.89 

7 

.978  7571 

31.09 

.989  9948 

3«34 

.001 

3272 

31.61 

.012 

7594 

31.90 

8 

.978  9436 

31.09 

.990  1829 

3>^35 

.001 

5169 

31.62 

.012 

9508 

31.90 

U 

.979   1302 

31.10 

.990  3710 

3«-3S 

.001 

7066 

31.62 

.013 

1422 

31.91 

!     10 

1.979  3168 

31.10 

1.990  5591 

31.36 

2. Co  I 

8963 

31-63 

2.013 

3337 

31.91 

11 

•979  S°U 

31. II 

■99°  7473 

3'^36 

.002 

0861 

31.63 

.013 

52^2 

31.92 

!     Vi 

.979  6901 

31.11 

.990  9355 

3>-37 

.002 

2759 

31.64 

.013 

7167 

31.92 

i;t 

.979  8768 

31.11 

•99'    «i37 

3«-37 

.002 

4658 
6557 

31.64 

.013 

9083 

31-93 

11 

.980  0635 

31.12 

.991    3119 

31.38 

.002 

31.65 

.014 

0999 

3193 

15 

1.980  2502 

31.12 

1.991   5002 

31.38 

2.002 

8456 

31.65 

2.014 

2915 

31-94 

10 

.980  4369 

31.13 

.991   6885 

31.38 

.003 

0355 

31.66 

.014 

4831 

3 '-94 

17 

.980  6237 

3'-i3 

.991   8768 

31.39 

.003 

2254 

31.66 

.014 

tJf^ 

3'-95 

18 

.980  8105 

31.13 

.992  0651 

3'-39 

.003 

4»54 

3i.r^7 

.014 

8665 

3'-9; 

19 

.980  9973 

31.14 

.992  2535 

31.40 

.003 

6054 

31.67 

.015 

0582 

31.96 

20 

1.981    1842 

31.14 

1.992  A419 
.992  6304 

31.40 

2.003 

7955 

31.68 

2.015 

2500 

31.96 

21 

.981   3710 

3i^»5 

31.41 

.003 

9855 

31.68 

.015 

4418 

3'-97 

22 

.981   5579 

3i-»S 

.992  8188 

31.41 

.004 

'I^o 

31.(58 

.015 

6336 

3' -97 

23 

.981   7449 

31.16 

.()()^   0073 

31.42 

.004 

3658 

31.69 

.015 

8255 

31.98 

24 

.981  9318 

31.16 

•993  '95« 

31.42 

.004 

5559 

31.69 

.016 

0174 

31.98 

25 

1.982   1188 

31.16 

1.993  38*3 

31.42 

2.004 

7461 

3'-70 

2.016 

2093 

31.99 

26 

.982   3058 

31.17 

•993  5729 

3i'43 

.004 

93''3 

31.70 

.016 

4012 

31.99 

27 

.982  4928 

3'-'7 

•993  7615 

3'-43 

.005 

1165 

31.71 

.016 

5932 

32.00 

28 

.982  6798 

31.18 

•993  9501 

3I-44 

.005 

3168 

31.71 

.016 

7852 

32.00 

29 

.982  8669 

31.18 

•994  '387 

3 1  44 

.005 

5071 

31.72 

.Oiu 

9772 

32.01 

1     30 

1.983  0540 

31.18 

1-99+  3274 

31-45 

2.005 

6974 

31.72 

2.017 

1693 

32.01 

31 

.983  24 1 1 

31.39 

•994  5'6i 

31-45 

.005 

8878 

3>-73 

.017 

3614 

32.02 

32 

.983  4283 

31.19 

•994  7048 

31.46 

.006 

0781 

3«-73 

.017 

5535 

32.02 

1     33 

.983  6155 

31.20 

■994  8936 

31.46 

.006 

2685 

3»-74 

.017 

7456 

32.03 

1    34 

.983  8027 

31.20 

•995  08*3 

31.46 

.006 

4590 

3'-74 

.017 

9378 

32^03 

!     35 

1.983  9899 

31.21 

1.995  2711 

31-47 

2.006 

6494 

31-75 

2.018 

1300 

32.04 

1     36 

.984  1772 

31.21 

.995  4600 

3'-47 

.006 

8399 

31-75 

.018 

3223 

32.04 

1    37 

.984  3644 

31.22 

.995  6488 

31.48 

.007 

0304  1 

31.76 

.018 

5'i5 

32.05 

1    38 

.984  5517 

31.22 

•995  8377 

31.48 

.007 

2210  1 

31.76 

.018 

7068 

32.05 

!    39 

.984  7391 

31.22 

.996  0266 

31.49 

.007 

41  '6   1 

3'-77 

.018 

8992 

32.06 

40 

1.984  9264 

31-13 

1.996  2155 

3«-49 

2.007 

6022 

3'-77 

2.019 

0915 

32.06 

41 

.985   1138 

3i-i3 

■')')"  4°45 

31.50 

.007 

792S   1 

31-77 

.019 

2839 

32.07 

42 

.985   3012 

31.24 

■996  5935 

31.50 

.007 

9835 

31.78 

.019 

4763 

32.07 

43 

.985   |886 

31.24 

.996  7825 

3'-5i 

.008 

1742  ' 

31.78 

.019 

6688 

32.08 

44 

.985  6761 

ii-H 

.996  9716 

31-51 

.008 

3649   1 

31-79 

.019 

8613 

32.08 

45 

1.985  8636 

31.25 

1.997   1606 

31.51 

2,oo8 

5556   i 

31-79 

2.020 

0538 

32.09 

4fi 

•"^o^  °5'5 

3i^2S 

•997  3497 

31.52 

.008 

7464 

31.80 

.020 

2463 

32.09 

47 

.986  2386 

31.26 

•997  5389 

3'^5i 

.008 

9372 

31.80 

.020 

4389 

3-"^ 

48 

.986  4262 

31.26 

.997  7280 

31-53 

.009 

1280  ! 

31.81 

.020 

6315 

32.10 

49 

.986  6138 

3i'i7 

•997  917a 

3»-53 

.C09 

3189  j 

31.81 

.020 

8241 

32.11 

50 

1.986  8014 

31.27 

1.998  1064 

31-54 

2.009 

5098  i 

31.82 

2.021 

0168 

31. 1 1 

51 

.986  9890 

31.28 

.998  2956 

31-54 

.009 

7007    ! 

31.82 

.021 

2095 

32.11 

52 

.987   1767 

31.28 

.998  4849 

31-55 

.009 

8917 

31.83 

.02  1 

4022 

32.12 

53 

.987  3644 

31.28 

.998  6742 

31-55 

.010 

0826 

31.83 

.021 

5949 

32.13 

54 

.987  5521 

31.29 

.998  8635 

31-56 

.010 

2736 

31.84 

.021 

7877    : 

32^i3 

55 

1.987  7398 

31.29 

1.999  0529 

31.56 

2.010 

4647    i 

31.8+ 

2.021 

9805    ' 

32.14 

56 

.987  9276 

31.30 

•999  2421 

31.56 

.010 

6557    1 

31.85 

.022 

'734 

32.14 

57 

.988  1154 

31.30 

•999  43 '6 

31-57 

.010 

8468 

31.85 

.022 

3662 

32.1,- 

58 

.988  3032 

31.31 

.999  621 1 

3«-57 

.oil 

0380 

31.86 

.022 

5591 

32.15 

59 

.988  49H 

31-31 

.999  8105 

31.58 

.011 

2291 

31.86 

.022 

7521 

32.16 

eo 

1.988  6789 

3I-3' 

2.000  0000 

31.58 

2,011 

4203 

31.87 

2.022 

9450 

3».i6 

in  a  Parabolic  Oibit. 


TABLE  VI. 

For  tliiding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


91^ 


loK  M. 


Diff.  1". 


2.01  I    4203     I 

.oil    6115    ' 

.on    8027    ! 
.011  9940  \ 

.012  1853  ; 

2.012  3766  1 
.012  5680  I 
.012  7594 
.012  9508 
.013  1422 

2.013  3337 
.013  52^2 
.013  7167 
.013  9083 
.014  0999 

291S 
483. 
6748 
8665 
0582 

2500 
4418 
6336 
8255 
0174 

2093 
4012 

5932 
7852 

9772 

1693 
3614 

55  35 
7456 
9378 

1300 
3223 

5145 
7068 

8992 

0915 
1839 
47^' 3 
6688 

8613 
0538 
2463 
43«9 
6315 
8241 

0168 
2095 
4022 
5949 

7877 


2.014 
014 
014 
1014 

,015 

2.015 
.015 
.015 
.015 
.016 

2.016 
.016 
.016 
.016 

•  Oku 

2.017 
.017 
.017 
.017 
.017 

2.01S 
.018 
.018 
.018 
.018 

2.019 
.019 
.019 
.019 
.019 

1.020 
.020 
.020 
.020 
.020 

2. 021 
.021 
.02  1 
.021 

.021 


3'-S7 
3i.8,>< 
31.88 
31.89 

31.89 
31.89 
31.90 
31.90 
31.91 

31.91 
31.92 
3'-9i 
3  •■93 
31-93 

3«94 
3«-94 
3>-95 
3 '9  5 
31.96 

31.96 
31-97 
3''97 
31.98 
31.98 

3'-99 

i    3 '99 
I   32.00 

i    31-0° 
I    32.01 

i 
32.01 

I   32.02 

'     32-Oi 

32.03 
32.03 

32.04 
32.04 
32.0^ 
31.0^ 
32.06 

32.06 
32.07 
32.0? 
31.08 

32.08 

32.09 
32.09 


^ 
5 
5 

6 
6 

2.021  9805  ' 
.022  1734 
,022  3662 
.022  5591 
.022  7521 

7 

2.022  9450 

l\ 

920 

93° 

94° 

95 

0       i 

let:  -M. 

iHfr.  1". 

1„K 

M. 

Dim  1". 

32.48 

2.046  3296 

Diff.  1". 

1  kM. 

Diff.  1". 

33. '5 

0 

1.022  9450 

32.16 

2.034 

5797 

32.80 

2.058  2005 

1 

.023  1380  ! 

32.17 

.034 

7745 

32.48 

.046  <;264 

32.81 

.058  3994 

33-' 5 

2 

.023  3311  j 

32.17 

.034 

9694 

32.49 

.046  7233 

32.82 

.058  59S3 

33.16 

3 

.023  5241  i 

32.18 

.035 

1644 

3249 

.046  9202 

32.82 

.058  7973 

33.6 

4 

.023  7172  1 

32.18 

.035 

3593 

32.50 

.047  1172  i 

32.83 

.058  9963 

33.17   i 

5 

2.023  9103  1 

32.19 

2.035 

5543 

32.50 

2.047  3141 

32.83 

2.059  1953 

33- « 8 

6 

.024  1035 

32.19 

•035 

7494 

32.51 

.047  5111 

32.84 

.059  3944 

33-'8 

7 

.024  2967 

32.20 

.035 

9444 

32.51 

.047  7082 

32.84 

•059  593  5 

33-'9 

8 

.024  4899  1 

32.20 

.036 

«395 

32.52 

.047  9053 

32.85 

.059  7927 

33. '9 

9 

.024  6831 

32.21 

.036 

3  347  , 

32.52 

.048  1024 

32.85 

.059  9919 

33.20 

10 

2.024  8764 

32.21 

2.036 

5298 

32-53 

2.048  2995 

32.86 

2.060  1911 

3321 

li 

.025  0697 

32.22 

.036 

7250 

32-53 

.048  4967 

32.87 

.060  3904 

33.21 

Vi 

.025  2630 

32.22 

.036 

9202 

32.54 

.048  6939 

32.87 

.060  5897 

33.22 

III 

.025  4564 

32.23 

.037 

1155 

3254 

.048  8912 

32.88 

.060  7890 

33.22 

14 

.025  6498 

32.23 

.037 

3108 

32.55 

.049  0884 

32.88 

.060  9884 

33.23 

15 

2.025  8432 

32.24 

2.037 

5061 

32-55 

2.049  2857 

32-89 

2.061  1878 

3324 

16 

.026  0367 

32.24 

.037 

7015 

32.56 

.049  4,(31 

32.89 

.061  3872 

33-24 

17 

.026  2301 

32.25 

.037 

8969 

32.57 

.049  6805 

32.90 

.061  5867 

3325 

18 

.026  4236 

32.26 

.038 

0923 

32-57 

.049  8879 

32.90 

.061  7862 

3325 

10 

.026  6172 

32.26 

.038 

2877 

32.58 

.050  0753 

32.91 

.061  9857 

33.26 

20 

2.026  8108 

32.27 

2.038 

4832 

32.58 

2.050  2728 

32.92 

2.062  1853 

33.27 

21 

.027  0044 

32.27 

.038 

6787 

32.59 

.050  4703 

32,92 

.062  3849 

33.27 

22 

.027  19S0 

32.28 

.038  8743 

32.59 

.050  6679 

32.93 

.062  5.X46 

33.28 

23 

.027  3917 

32  28 

.039 

0699 

32.60 

.050  8655 

32.93 

.062  7842 

33-28 

24 

.027  5854 

32.29 

.039 

2655 

32.61 

.051  0631 

32.94 

.062  9840 

33-29 

25 

2.027  7791 

32.29 

2.030 

461 1 

32.61 

2.051  260S 

32-95 

2.063  1S3-/ 

33-3° 

20 

.027  9729 

3-3° 

.039 

6568 

32.62 

.051  45S5 

32.95 

''3  3835 

33-30 

27 

.028  1667 

3i'3° 

.039 

8525 

32.62 

.c'l  6562 

32.96 

.063  5S3; 

33-3' 

28 

.028 ,3605 

32-3« 

.040 

0482 

32.63 

.051  8539 

32.96 

.063  7832 

33-3' 

2» 

.028  5544 

32.31 

.040 

2440 

32.63 

.052  0517 

32.97 

.063  9831 

33.32 

:io 

2.028  7483 

32.32 

2.040 

4399 

32.64 

2.052  2496 

32.97 

2.064  1 83 1 

33-33 

:u 

.028  9422 

32.32 

.040 

6357 

32.64 

.052  4474 

32.98 

.064  3830 

33-33 

;j2 

.029  1361 

3*-33 

.040 

8316 

32.65 

.052  6453 

32.98 

.064  5830 

33-34 

33 

.029  3301 

3^-33 

.041 

0275 

32.65 

.052  8432 

32.99 

.064  7S31 

33-34 

34 

35 

.029  5241 
2.029  7182 

3»-34 
32.3.: 

.041 

2.04; 

2234 
4194 

32.66 
32.67 

.053  0412 
2053  239-. 

33.00 
3  3-00 

.064  9832 

33-35 

2.065  '833 

33.36 

30 

.029  9123 

32.35 

.041 

6154 

32.67 

.053  4372 

33-01 

.065  3834 

33.36 

37 

.030  1064 

31-35 

.041 

8,14 

32.68 

.053  6353 

33.01 

.065  5836 

33  37 

38 

.030  3005 

32.36 

.042 

0075 

32.68 

•053  8334 

33.02 

.065  7839 

33  37  ■ 

39 

.030  4947 

32.36 

.042 

2036 

32.69 

.054  0315 

33.03 

.065  9841 

33-38 

10 

2.030  6889 

32.37 

2.042 

3998 

32.69 

2.054  2297 

33-03 

2.ob6  1844 

3^39 

4« 

.030  S831 

32-37 

.042 

5960 

32.70 

.054  4279 

33.04 

.066  3847 

53-39 

12 

.031  0774 

32.38 

.042 

7922 

52.70 

.054  6262 

33.04 

.066  5851 

33-40 

13 

.031  2717 

32-39 

.042 

98x4 

3  ■■.7 1 

.054  8244 

33.05 

.066  78155 

33-40 

14 

.031  4660 

32-39 

.043 

1847 

32.71 

.055  0227 

33-05 

.066  9860 

33-4' 

45 

2.03 1  6604 

32.40 

2.043 

3810 

32.72 

2.055  2211 

33.06 

2.067  '865 

33-42 

10 

.031  8548 

32.40 

.04-; 

5773 

32-73 

.055  4195 

33.07 

.067  3S70 

33-42 

17 

.032  0492 

32.41 

.043 

7737 

32-73 

.055  6179 

33.07 

.067  5875 

33.43 

48 

.032  2437 

32-4> 

-043 

9701 

32.74 

.055  8163 

33.08 

.067  7881 

33-43   ; 

49 

.032  438'2 

32.42 

.044 

1665 

32.74 

.056  0148 

33-08 

.067  9887 

33-44 

.50 

2.032  6327 

32-42 

2.044 

3630 

32.75 

2.056  2133 

33-09 

2.068  1894 

33.45 

51 

.032  8272 

32-43 

.044 

5595 

32-75 

.056  4119 

33.10 

.068  3901 

33.45 

52 

.033  0218 

32-43 

.044 

7561 

32.76 

.056  6105 

33.10 

.068  5908 

33.46 

53 

.033  2164 

32.44 

.044 

9=;26 

32.76 

.056  8091 

33.11 

.c68  7916 

33-47 

51 

^33  4HI 

32-44 

.045 

'492 

32.77 

.057  0078 

33.11 

.068  9924 

33-47 

55 

2.033  6058 

32-45 

2.045 

3459 

32.78 

2.057  2065 

33.12 

2.069  1933 

33.48 

50 

.033  8005 

32.45 

.045 

5426 

32-78 

.057  4052 

33.12 

.069  3942 

33-48  i 

57 

.033  9952 

32.46 

.045 

7393 

32.79 

.057  6040 

33->3 

.069  5951 

33.49 

58 

.034  1900 

32.47 

.045 

9360 

32.79 

.057  8028 

33.'4 

.069  7960 

33-50  i 

59 

.034  3848 

32-47 

.046 

1328 

32.80 

.058  0016 

33.'4 

.069  9970 

33-50  I 

60 

2.034  5797 

32.48 

2.046 

3296 

32.80 

2.058  2005 

33.'S 

2.070  1980 

33-51 

-- 

589 

TABLE  VI. 

For  finding  the  True  Anomaly  or  llie  Tinu-  from  tlie  Perihelion  in  a  Parabolic  Orbit. 


V. 

0 

96 

0 

,97 

0 

98 

0 

99 

0 

lo^ 

M. 

IHff.  I". 

33-5« 

logM. 

Diff.  1". 

los  M. 

Diir.  1". 

logM. 
2. 1 07   0109 

Diir.  1". 
34.69 

2.070 

1980 

2.0S2 

3181 

33.88 

2.094  5971 

34.28 

1 

.070 

399 « 

33o« 

.082 

53'6 

33.89 

.094  8028 

34.29 

.107   2190 

34.70 

•z 

.070 

6002 

33'52 

.082 

7349 

33  90 

.095   0085 

34-29 

.107   4171 

34-70 

3 

.070 

8014 

33-53 

.082 

9  3 '"'3 

33.90 

-095   2143 

34-30 

.107   6355 

34-71 

4 

.071 

0025 

33-53 

.083 

1418 

33.91 

.095   4201 

34-31 

.107   8437 

34-72 

5 

1.071 

2037 

33-54 

1.083 

5+53 

33-91 

2  095   6260 

34- 3' 

2.108   0521 

34-72 

6 

.071 

40,-0 

33-54 

.0S3 

5488 

3392 

.091;   8318 

3432 

.108    2604 

34-73 

7 

.071 

6063 

33-55 

.0S3 

7523 

33-93 

.096  0378 

34-33 

.108   46S9 

3-V74 

8 

.071 

8076 

33-5*' 

.083 

9559 

33-94 

.096   2438 

3433 

.108   6773 

34-'' 5 

0 

.072 

0090 

33-56 

.084 

1596 

33-94 

.096  4498 

j4-34 

.108   8858 

34-75 

10 

2.072 

2104 

33-57 

2.0S4  3633 

33-95 

1  096  6558 

34-35 

2.109  0944 

34-76 

11 

.072 

4118 

33-58 

.084   5670 

33-96 

.096   8619 

34-35 

.109   3029 

34-77 

12 

.072 

6133 

3358 

.084 

7707 

33.96 

.097  06S1 

34.36 

.109   5  1 16 

34-77 

i    i;i 

.072 

8148 

33-59 

.084 

9745 

33-97 

-=97   2742 

34-37 

.109   7201 

34-78 

14 

.073 

0163 

33-59 

.085 

'783 

33-98 

.097  4804 

34-37 

.109   9289 

34-79 

15 

2.073 

2179 

33.60 

2.0S5 

^Sr- 

33.98 

2.097  6867 

34-38 

2.110   1377 

34-80     , 

16 

.073 

4195 

33-61 

.08  s 

S86i 

33-99 

.097   8930 

34-39 

.110   3465 

34-80 

17 

.073 

6212 

33-61 

.08  i; 

7901 

33-99 

.09X    0993    ^ 

34-39 

-«io   5553 

34-81 

18 

.073 

8229 

33.62 

.085 

994' 

34.00 

.098    3057 

34-40 

.110  7642 

34-82 

lU 

.074 

0246 

33-63 

.086 

1981 

34.01 

.098    51 21 

34-4» 

.110  9731 

34.82 

20 

2.074 

2264 

33-63 

2.086 

4021 

34.01 

2.098    7186 

34-4« 

2. Ml     1821 

34-83 

21 

.074 

4282 

33-64 

.086 

6062 

3-V.02 

.098    9251 

34-42 

.111     39II 

34-84 

22 

.074 

6301 

3364 

.086 

8104 

34-03 

.099    I  316 

34-43 

.111    6001 

34-85 

23 

.074 

8320 

33-65 

.087 

0146 

34-03 

.099    3382 

34-43 

.111     8091 

34-85 

24 

.075 

0339 

33.66 

.087 

2188 

34.04 

.099    5449 

34-44 

.112    0184 

34.86 

25 

2.075 

2358 

33.66 

2.C'^7 

4231 

34-05 

2.099    7515 

34-45 

2.112    2275 

34-87 

26 

.075 

4378 

33-67 

.oXy 

6274 

34-05 

.099    9581 

34-45 

.112    4368 

34-87     ■ 

27 

.075 

6399 

33.67 

.087 

83,7 

34.06 

.100    1650 

34.46 

.112    6460 

34-88     : 

28 

.075 

84.9 

33.68 

.08  S 

0361 

34-07 

.100   3718 

34-47 

.112    8553 

34-89     ' 

29 

.076 

0440 

33-69 

.08  .S 

2405 

34-07 

.100    5786 

34-48 

.113    0647 

34-90 

30 

2.076 

2462 

3369 

1.088 

4449 

3408 

2.100    7855 

3448 

2.113    2741 

34-?'> 

31 

.076 

4484 

33-70 

.0S8 

6494 

3409 

.100    9914 

34-49 

.113    4835 

34-91 

32 

.076 

6507 

33-71 

.08  8 

8540 

34.09 

.101     1993 

34-50 

.113    6930 

34-'.2 

33 

.076 

8529 

33-71 

.089 

0586 

34.10 

.101    4063 

34-50 

.113    9025 

34-92 

34 

.077 

0552 

33-71 

.089 

2632 

34.11 

.101    6134 

34-51 

.114    1121 

34-y3 

35 

2.077 

*57S 

33-73 

2.0S9 

4678 

34.11 

2.IOI     8204 

34-52 

2.114    3*17 

34-94     \ 

30 

.077 

4599 

33-73 

.0S9 

6725 

34.12 

.102    0276 

3452 

.114    5313 

34-95      i 

37 

.077 

6623 

33-74 

.089 

8772 

3412 

.102    2347 

34-53 

.114    7410 

34-95     , 

38 

.077 

8647 

33-74 

.090 

0820 

34- « 3 

.101  4419 

34-54 

.114    9508 

34-96   : 

39 

.078  0672 

33-75 

.090 

2868 

34.14 

.102    6492 

34-54 

.115     1605 

34-97 

40 

i.078 

2697 

33.76 

2.090 

4917 

34->5 

2.102    8564 

34-55 

2.115     3704 

34-97 

41 

.078 

47  •4  3 

33-76 

.090 

6966 

34-15 

.103    0638 

34-56 

.115     5802 

34-98 

42 

.078  6749 

33-77 

.090 

9015 

34.16 

.103    27II 

34-56 

.115    7901 

34-99 

43 

.078  8775 

33.78 

.091 

1065 

34-17 

.103    47X5 

34-57 

.1 16   OOCl 

35.00 

44 

.079 

0802 

33.78 

.091 

3115 

34- « 7 

.103    6860 

34.58 

.116  2101 

35.00 

45 

2.079 

2829 

33-79 

1.091 

5165 

34.18 

2.103    893s 

34-59 

2.116  4201 

35-01 

46 

.079 

4857 

33.80 

.091 

7216 

34-19 

.104    lOIO 

34-59 

.1 16  6301 

35.02        ! 

47 

.079 

6885 

33.80 

.091 

9268 

34.19 

.104  3086 

34.60 

.ii'S  8403 

35.01 

48 

.079 

8913 

33-81 

.092 

1319 

34.20 

.104  5162 

34.61 

.117  0505 

35-03 

49 

.080 

0942 

33. M 

.092 

3371 

34.20 

-'04  7239 

14.61 

.117  2607 

35-04 

50 

7.  080 

2971 

3382 

2.092 

5444 

34.21 

2.104  93 '6 

34.62 

2.117  4710 

35-05 

51 

.080 

5000 

3383 

.092 

7477 

3422 

.105   1393 

3463 

.117  6813 

3505 

1     52 

.080 

7030 

33,83 

.092 

953° 

34.22 

.105  3471 

34-63 

.117  8916 

35.00 

53 

.080 

9060 

33-84 

.093 

1584 

34-23 

.lOf     5549 

34.64 

.118     1020 

35-07 

54 

.081 

1091 

33-85 

.093 

3638 

34-24 

.105  7628 

34.65 

.118    3124 

35.08      ^ 

55 

2.081 

3122 

33-85 

1.093 

5692 

34-»5 

2.105  9707 

34.66 

2.118    5229 

35.08      . 

50 

.081 

5153 

33-86 

.093 

7747 

34-25 

.106   1786 

34-66 

.118    7334 

35-09     ; 

57 

.081 

7185 

33-87 

.093 

9^°3 

34.26 

.106  3866 

3467 

.118    9440 

35-'o     ! 

58 

.081 

9217 

"•^Z 

.094 

1858 

3427 

.106  5947 

34.68 

.119    1546 

35-'o 

59 

.082 

1249 

33.88 

.094 

3914 

3427 

.106  8027 

34.68 

.119    36^2 

35-" 

60 

2.082 

3282 

33-88 

2.094 

5971 

34.28 

2.107  °io9 

34.69 

1.119  5759 

35.U 

590 

mA 


ibolie  Orbit. 


TABLE  VI. 

For  finding  the  True  Anomaly  or  tiie  Time  from  the  Periiielion  in  a  Parabolic  Orbit. 


99^ 


3217 

53'3 

7410 
9508 
1605 

3704 
5802 
7901 
con 
2101 

4201 
6301 


Diff.  1". 

34.69 

34.70 
3470 
34-71 
34-72 

34-72 
34-7) 
34-74 
34- ■'5 
34-7i 

34.76 

34-77 
34-77 
34. 7  S 

34-79 

34.80 

34, So 
34.S1 
34.S2 
34.S2 

34-5=3 
34->*4 
34-i*5 
34.i<5 
34.86 

54-i<7 
34-!<7 
34.88 
34.89 
34.90 

34-9'' 
34-91 
34-'72 
34-'^- 
34-'^) 

34-94 
3495 
349i 
34-'''' 
34-97 

34-9'' 
34-9*f 
34-99 

3vOO 
35.00 

35.01 
35.01 


3^- 


*^403 

0505      3v03 

2607       35.04 


4710 
6813 
8916 

1020 
3124 

5229 

7334 
9440 
1546 
36-2 

5759 


35-°5 

35-°i 
3voO 

35.08 
35.08 

35-°9 
35.10 

35''° 
35" 
35.12 


l\ 


O 
1 

a 
:t 
4 

5 

0 

8 
» 

10 
II 

vz 
1:} 
11 

15 
1(> 
17 

18 
19 

-zo 

•il 
Ti 
23 

24 

25 
2(S 
27 

28 
21) 

30 
31 
32 
33 
31 

35 

3U 
37 

38 
3« 

10 
II 
12 
43 
41 

45 
40 
47 

48 
4U 

50 
51 
52 
53 
54 

55 
5(i 
57 
58 
5» 

00 


100° 

101° 

loR  M. 

Dlff.  1". 

logM. 

Diir.  1". 

.119  5759 

35.12 

2.132  2989 

35-57 

.119  7867 

35-'3 

.132  5123 

35-57 

.119  9974 

35'3 

.132  7258 

35-58 

.120  2083 

35-'4 

.132  9393 

35-59 

.120  4191 

35-15 

-'33  '529 

35.60 

.120  6301 

35.16 

S.I33  3665 

35.61 

.120  8410 

35.16 

.133  5802 

35.61 

.121  0520 

35-'7 

•133  7939 

35.62 

.121  2630 

35.18 

.134  0076 

35-63 

.121  4741 

35'9 

.134  2214 

35-64 

.121  6853 

35-'9 

J. 134  4352 

35-64 

.121  8965 

35.20 

-'34  649' 

35-65 

.122  1077 

35.21 

.134  8631 

35.66 

.122  3190 

35.21 

-'35  0770 

35-67 

.122  5303 

35.22 

.135  2910 

35-67 

..122  7416 

35-23 

2.135  5051 

35-68 

.122  9530 

35-24 

.135  7192 

35-69 

.123  1644 

35-24 

-'(5  93  34 

35-70 

•123  3759 

35-25 

.i}6  1476 

35-7' 

.123  587s 

35.26 

.136  3619 

3i7i 

.123  7990 

35-27 

2.^36  5762 

35-72 

.124  0107 

35-27 

.136  7905 

35-73 

.124  2223 

35.28 

-137  0049 

35-74 

.124  4340 

35-9 

.137  2193 

35-74 

.124  6458 

35.30 

-'37  4338 

35-75 

.124  8576 

35-30 

2.137  6484 

35-76 

.125  0694 

35-3' 

.137  8630 

35-77 

.125  2813 

35-32 

.138  0776 

35-77 

.125  4933 

35-33 

.138  2922 

35-78 

.125  7052 

35-33 

.138  5070 

35-79 

.125  9173 

35-34 

2.138  7217 

35.80 

.126  1293 

35-35 

.138  9365 

35-81 

.126  341^ 
.126  5536 

35-35 

.139  1514 

35.81 

35-36 

•139  3663 

35-82 

.126  7658 

35-37 

-139  5813 

35-83 

..126  9780 

35-38 

2.139  7963 

35-84 

.127  1903 

35-39 

.140  0113 

35.84 

.127  A027 
.127  6151 

35-39 

.140  2264 

35-85 

35.40 

.140  4415 

35-86 

.127  8275 

35-4' 

.140  6567 

35-87 

.128  0400 

35.42 

2.140  8720 

35-88 

.128  2525 

35.42 

.141  0873 

35-88 

.128  4650 

35-43 

.141  3026 

35.89 

.128  6776 

35-44 

.141  5180 

35-90 

.128  8903 

35-45 

.141  7334 

35-9' 

.129  1030 

35-45 

2.141  9A89 
.142  1644 

35.92 

.129  3157 

35.46 

35.92 

.129  5285 

35-47 

.142  3799 

35-93 

.129  7414 

35.48 

.142  5955 

35-94 

.129  9542 

35-48 

.142  81 12 

35-95 

L.130  1672 

35-49 

2.143  0269 

35-96 

.130  3801 

35-50 

•  '43  2427 

35-96 

.130  5931 
.130  8062 

35-5' 

.143  4585 

35-97 

35-51 

.143  6743 

35-98 

.131  0193 

35-52 

.143  8902 

35-99 

.131  2325 

35-53 

2.144  '062 

36.00 

.131  A457 

35-54 

.144  3222 

36.00 

.131  6589 

35-54 

.144  5382 

36.01 

.131  8722 

35-55 
35-56 

•144  7543 

36.02 

.132  0855 

•'44  9704 

36.03 

..132  2989 

35-57 

2.145  '866 

36.03 

102 

0 

IobM. 

Diff.  1". 

36.03 
36.04 
36.05 
^6.06 
35.07 

.145  1866 
.145  4028 
.145  6191 

-'45  8354 
.146  0518 

.146  2682 
.146  4847 

.146  70.' 2 

.146  9 1/8 
-'47  '344 

36.07 
36.08 
36.09 
36.10 
36.11 

-'47  35'o 
-'47  5677 
•'47  7845 
.148  0013 
.148  2182 

36.11 
36.12 

36.13 
36.14 

36-15 

.148  4351 
.148  6520 
.148  8690 
.149  oS6i 
.1.^9  3032 

36.17 
36.18 
36.19 

-'49  5203 
-'49  7375 
-'49  9547 
.150  1720 
.150  3893 

36.19 
36.20 
36.21 
36.22 
36.23 

.150  6067 
.150  8242 
.151  0417 
.151  2592 
.151  4768 

36-23 
36.24 

36.26 
36-27 

.151  6944 
.151  9121 
.152  1298 
.152  3476 
.152  5654 

36.28 
36,28 
36.29 
36.30 
36.31 

-152  7833 
.153  0012 
.153  2192 
-'53  4372 
-153  6552 

36.32 
36.32 
36.33 
36-34 
36.35 

-'53  8734 
.154  0915 
.154  3097 
.154  5280 
.154  7463 

5^3| 
36.36 

36.37 
36.38 

36.39 

•  154  9647 

•  155  '831 
.155  4015 
.155  6200 
.155  8386 

36.40 
36.41 
36.41 
36.42 
36.43 

.156  0572 
.156  2759 
.156  4946 
.156  7133 
.156  9321 

36.44 
36.45 

^^^^ 
36.46 

36.47 

■'57  i5'o 
•'57  3699 
•'57  5889 
.157  8079 
.158  0269 

36.48 
36.49 
36.50 
36.50 
36.51 

.158  2460 

36.52 

103^ 


log 

M.    1 

! 

.158 

2460 

.158 

4652 

.158 

6844 

.158 

9030  , 

.159 

1229  : 

■'59 

3423  i 

-'59 

5617  1 

.159 

7811  i 

.160 

0006  ; 

.160 

2  202 

.160 

4398 

160 

6594  ; 

.160 

8791  1 

.161 

09X9  1 

.ibi 

3187 

.161 

5385 

.161 

75*>4  i 

.161 

9784  ! 

.162 

,984  ' 

.162 

41X5  1 

.162 

6386 

.162 

8587 

.16-! 

0789  ! 

.163 

2992  i 

.163 

5 '95  1 

.163 

7398 

..63 

9602  1 

.164 

1807  j 

.164 

4012  1 

.164 

6218  I 

.164  8424  1 

.165  0630  i 

.165 

2837  1 

.,65 

5045 

.165 

7253 

.165 

9462 

.166 

1671 

.166 

3881 

.166 

6091 

.166 

8301 

.167 

0513 

.167 

2724 

.167  4936  1 

.167 

7149 

.167 

9362 

.168 

1576 

.168 

3790 

.168 

6005 

.168 

8220 

.169  0436  1 

.169 

2652 

.169  4869 

.169 

7087 

.-69 

9304 

.170 

1523 

.170 

37-J2 
5961 

.170 

.170 

8181 

.171 

0401 
2622 

.171 

.171 

4844 

nifl.  I". 

36.52 

36.53 
36- 54 
36.55 
36.55 

36.56 

36.57 
36.58 

36.59 
36.60 

36.60 
36.61 
36.62 
36.63 
36.64 

36.65 
36.65 
36.66 
36.67 
36.68 

36.69 
36.70 
36.70 
36.71 
36.72 

36.73 
36.74 
36.74 

36.75 
36.76 

36.77 

36.78 
36.79 
36. Xo 
36.81 

36.81 
36.82 

36.83 
36.84 
36.85 

36.86 
36.87 

36.87 
36.88 
36.89 

36.90 
36.91 
36.92 

3693 
36.93 

36.94 
36.95 
36.90 
36.97 
36.98 

36.99 
36.99 

37.00 
37.01 
37.02 

37.03 


591 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


|7 


V. 

1 

104° 

,  105° 

106° 

107° 

log 

M. 

Diff.  1". 

log  M. 

Diir.  1". 
37-56 

logM. 
2.198    5282 

Diir.  1". 

log  M. 

Diff.  1".    ! 

38.68 

0 

2.I7I 

4844    1 

37-03 

2.184  9092 

38.11 

2.212    3493 

1 

.171 

7066 

37.04 

.185    1346 

37^57 

.198   7568 

38.12 

.212    5814 

38.69  ; 

2 

.171 

9288 

37^05 

.185    3600 

37^57 

.198   9856 

38-'3 

.212    8136 

38.70   i 

3 

.I7Z 

5" 

37-05 

-'85    5855 

37^58 

.199   2144 

38.14 

.213    0458 

38.7.  1 

4 

.172 

i73S 

37.06 

.185    8110 

37^59 

-'99  4432 

38.14 

.213    2781 

38.72  1 

5 

2.171 

P^"^ 

37.07 

2.186   0366 

37.60 

2.199  6721 

38-'5 

2.213    5104 

38^73 

1        6 

.172 

8184 

37.08 

.186   2622 

37^6i 

.199  9010 

38.16 

.213    7428 

3^-74     . 

i       7 

•»73 

0409 

37-09 

.i?6  4879 

37.62 

.200  1 300 

3!''Z 

•2'3  9753 

38-75 

1       ^ 

.173 

2634 

37.10 

.186  7137 

37^63 

.200  3591 

38.18 

.214  2078 

38.76 

9 

.173 

4860 

37.11 

•'86  9395 

37.64 

.200  5882 

38.19 

•214  4404 

3'<^77     : 

10 

2.173 

7087 

37.12 

2.187   1653 

37-65 

2. zoo  8174 

38.20 

2.214  6730 

38.78 

11 

•'73 

93'4 

37^ia 

.187  3912 

37-66 

.201    0467 

38.21 

.214  9057 

3S.79 

12 

.174 

1542 

37-13 

.187  6172 

37-67 

.201    2760 

38.22 

.215  1385 

3S.80 

13 

.174 

3770 

37^'4 

.187  8432 

37-67 

.201    5053 

38.23 

-2'5   37'3 

38.81 

14 

.174 

5999 

37^i5 

.188  0693 

37-68 

.201     7347 

38.24 

.215  6042 

38.82 

15 

2.174 

8228 

37.16 

2.188   2954 

37.69 

2.201     9642 

38.25 

2.215  8371 

38.83  ! 

16 

•«75 

0458 

37^i7 

.188   5216 

37-70 

.202    1937 

38.26 

.216  0701 

38.84 

17 

•175 

2688 

37.18 

.188  7478 

37-71 

.202    4233 

38.27 

.216   3032 

38.S5 

18 

•«75 

49'9 

37^i8 

.188  9741 

37-72 

.202    6529 

38.28 

.216  5363 

3S.X6  - 

19 

••75 

7150 

37^i9 

.189  2005 

37-73 

.202    8826 

38.29 

.216  7694 

38.87  . 

20 

^•>7S 

9382 
1615 

37.20 

2.189  4269 
.189  6533 

37-74 

2.203     1123 

38.30 

2.217  0027 

38.88 

21 

.176 

37.21 

37^75 

.203     3421 

38.31 

.217  2360 

38.?i 

22 

.176   384X 

37.22 

.189  8798 

37^76 

.203     5720 

38.31 

.217  4693 

3S.90 

23 

.176 

6081 

37.23 

.190  1064 

37-77 

.203     8019 

38.32 

.217  7027 

3S.91 

24 

.176 

83'5 

37-24 

.190  3330 

37-77 

.204    0319 

38.33 

.217  9362 

3S.92 

25 

2.177 

0550 

37-25 

2.190  5597 

37^78 

2.204    2619 

38.34 

2.21 8    2697 

38.93 

2« 

•'77 

2785 

37-25 

.190  7864 

37^79 

.204    4920 

38-35 

.218    4033 

3S.94 

2T 

•'77 

5020 

37.26 

.191   0132 

37-80 

.204    7222 

38.36 

.218    6369 

3S.95 

28 

•'77 

7256 

37.27 

.191   2401 

37-81 

.204    9524 

38-37 

.218    8706 

38.96 

•^9 

•'77 

9493 

37.28 

.191  4670 

37-82 

.205     1826 

38-38 

.219    1044 

38-97     , 

30 

2.178 

1730 

37.29 

2.191   6939 

37-83 

2.205    4129 
-205    6433 

38-39 

2.219    3382 

38.98 

31 

.178 

3968 

3730 

.191   9209 

37.84 

38.40 

.219    5721 

3S.99 

32 

.178 

6206 

37-3' 

,192   1480 

37-85 

.205     8737 

38.41 

.219  8o6i 

39.00 

33 

.178   844s 

37-32 

.192   3751 

37-86 

.206     1042 

38.42 

.220  0401 

39.01 

34 

•'79 

0684 

37-33 

.192  6023 

37-87 

.206    3348 

38.43 

.220  2741 

39.02 

35 

2  173 

2924 

37-33 

2.192  8295 

37.88 

2.206    5654 

38.44 

2.220  5082 

3y-°3   , 

36 

.179 

5164 

37-34 

.193  0568 

37.88 

.206    7961 

38^45 

.220  7424 

39-°4 

37 

•'79 

7405 

37-35 

.193  2841 

37.89 

.207    0268 

38.46 

.220  g7'57 

39-°5  ; 

38 

.179 

9646 

37-36 

-'93  5"5 

37.90 

.207    2575 

38.47 

.221   2110 

39.06 

39 

.180 

1S88 

37-37 

•'93  7389 

37.91 

.207    4884 

38.48 

.221  4453 

39.07 

40 

2.i8o 

4'3i 

37-38 

2.193  9664 

37.92 

2.207    7193 

38.49 

2.221   6797 

39.08     : 

41 

.180 

6374 

37-39 

.194  1940 

37^93 

.207    9502 

38.50 

.221  9142 

39-09     , 

42 

.180 

^^J7 

37.40 

.194  4216 

37-94 

.208     1812 

38-51 

.222   14X8 

39.10 

43 

.181 

0861 

37-41 

-194  6493 

37-95 

.208    4123 
.208    6434 

38.52 

.222   3834 

39.11    ; 

44 

.181 

3106 

37^4' 

.194  8770 

37-96 

38.53 

.222  6180 

39.12 

45 

2.181 

5351 

37.42 

2.195  1048 

37-97 

2.208    8746 

38.54 

2.222  8528 

39-'3 

46 

.181 

7597 

37^43 

.195  3326 

37.98 

.209     1058 

38-54 

.223  0876 

39.14 

47 

.181 

9S43 

37-44 

.195  5605 

37-99 

.209     3371 

38-55 

.223   3224 

3'>'5 

48 

.182 

1089 

37-45 

-'95  7885 

38.00 

.209    5685 

38.56 

•223   5573 

39.16 

49 

.182 

4337 

37.46 

.196  C165 

38.00 

-209  7999 

38-57 

.223  7923 

3'>'7 

50 

2.182 

6584 

37^47 

2.196  2445 

38.01 

2.21c  0314 

38-58 

2.224  0273 

39.18 

51 

.182 

8833 
1082 

37.48 

.196  4726 

38.02 

.210  2629 

38-59 

.224  2624 

3J-'9 

52 

.183 

3  7 '49 

.196  7008 

38.03 

.210  4945 

38.60 

.224  4975 

39.10 

53 

.183 

3331 

37-49 

.196  9290 

38.04 

.210  726t 

38.61 

-224  7327 

39.21 

54 

•'S3 

5581 

37-50 

-'97  '573 

38.05 

.210  9578 

38.62 

.224  9680 

39.22 

55 

2.183 

7831 

37-51 

2.197  3856 
.197  6140 

38.06 

2.21  I     1896 

38.63 

2.225  2033 

39.23 

56 

.184 

0082 

37.52 

^l-°l 

.211    4214 
.211    6533 

38.64 

•225  4387 

39.24 

57 

.184 

^2U 
45X6 

37-53 

.197  8425 

38.08 

38-65 

.225  6741 

3925 

58 

.184 

37-54 

.198  0710 

38.09 

.211     8852 

38.66 

.225  9096 

39.26 

59 

.184  6839 

37-55 

.198  2995 

38.10 

.212    1172 

38.67 

.226   1452 

39.27 

60 

2.184 

9092 

37-56 

2.198  5282 

38.11 

2.212    3493 

38.68 

».226    3808 

39.28 

592 

' 

bolic  Orbit. 


TABLE  VL 

For  finding  the  True  Anomaly  or  the  Time  from  the  Periiielion  in  a  Parabolic  Orbit. 


107= 


M. 


Diff.  1" 


3493   1 

38.M 

5814   1    38. 69 

8136    I    3S.70 

0458       38.71     1 

2781 

3X.72     1 

5104 

38-73 

7428 

3i<-74 

9753 

38-75 

2078 

38.76 

4404 

38-77 

6730 

38.78 

9057 

38-79 

13X5 

38.80 

3713 

38.x, 

6042 

38.82 

8371 

38-83 

0701       38. X4 

3032  !   38. S5 

>  53<'3   1   i'^-^(> 

'    7694     1     T,^'^! 

J  0027  1  38.88 

J    2360    !     jS.^'l 

7  4693 

38-90 

7  70^7 

38-91 

7  936^ 

3S.92 

8  3697 

38.93 

X  4033  1   3S.94 

8  6369   1   38.95 

8  8706 

3S.96 

9  1044 

38.97 

9  338* 

38.98 

9  5721 

38.99 

9  8061 

39.00 

0  0401 

39.01 

0  2741 

39.02 

0  5082 

39-°3 

0  7424 

39.04 

0  97'i7 

39.05 

I     21IO 

39.06 

I  4453 

39.07 

I  6797 

39.08 

I  9142 

39.09 

2   1488   1   39-10 

2   3>'34  i    39-" 

2  6180  i   39.12 

2  8528   :  39.13 

3  0876  ;  39.14 

3   3124  1   39-'5 

3  5573 

1   39. 11) 

3  79^3 

1   39->7 

4  0273  !  39.18 

4  2624  1   3;.i9 

4  4975    '   39-^-° 

4  73^7   i   39'2i 

4  9680   -    39.12 

5  2033   1   39-23 

5  4387   ;    39-^4 

5  6741    '    39  ^-5 

5  9096   '    39-26 

6   1452   .   39-27 

6  3808      3928 

V. 


o 
1 

'Z 

:i 
4 

5 
0 

7 

8 
9 

10 
II 

la 

14 

ir> 
10 

17 

18 
19 

20 
21 
22 
23 
24 

25 

20 

27 
28 
29 

30 
31 
32 
33 
34 

35 
36 
37 

38 
39 

40 
41 
42 
43 
44 

45 
40 

47 
48 
49 

50 
51 
52 
53 
54 

55 
5» 
57 
5H 
59 

60 


108^ 


logM.         Difr.  1". 


2.234 
.234 

-235 
.235 

•^35 

2.235 
.236 
.236 
.236 
.236 

2.237 
•^37 
•237 
■*37 
.23S 

2.238 
.238 
.238 
.238 
.239 

2.239 

•^39 
.239 
.240 
.240 


3808  I 
6165 

8523  ; 

0881  : 
3240    j 

5599  ! 
7959  ; 
0320  i 
2681  j 
5043 
7405 

9768  : 

2131  I 

4496  I 
6861 

9226      I 

159a  ! 

3959  1 
6326  i 
8694  ■ 

1063 

343a 
580Z 
8172 
0543 

2.232  2915 
232  5287 

232  7660 

233  0033 
233  2407 

2.233  4782 
.233  7157 

-a33  9533 
.234  1910 
.234  4287 


2.226 
.226 
.226 

.227 

.227 

2.227 
.227 
.228 
.228 
.228 

2.228 
.228 
.229 
.229 
.229 

2.229 
.230 
.230 
.230 
.230 

2.231 
231 
231 
231 
232 


6665 
9043 
1422 
3802 
6183 

8563 
0945 
3327 
5710 
8093 

0478 
2862 

5247 
7633 
0020 

2407 

4795 
7284 

9573 
1962 

4353 
6744 

9235 
1528 
3921 


2.240  6314 


39.28 
39.29 
39.30 

39-3« 
39.32 

39-33 
39-34 
39-35 
39.36 

39-37 

39-38 

39-39 
39.40 
39.41 
39.4a 

39-43 
39-44 

39-45 
39.46 

39-47 

39.48 

39-49 
39.50 

39-5« 
39-5a 

39-53 
39-54 
39-55 
39-56 
39-57 
39.58 

39-59 
39.60 
39.61 
39.63 

39.64 

39-65 
39.66 
39.67 
39.68 

39.69 
39.70 

39-71 
39.72 

39-73 

39-74 
39-75 
39.76 

39-77 
39.78 

39-79 
39.80 
39.81 
39.82 
39.83 

39.84 
39.8^ 
39.87 
39.8X 
39.89 

39.90 


2.240 
240 
241 
241 
241 


109= 


logH. 


6314 

8708 
1 103 
3498 
5894 


Dim  1". 


a. 241  8291 
,242  0688 
.242  3086 
.242  5485 
.242  7884 

2.243  0284 
.243  2685 
.243  5086 

•243  7488 
.243  9890 


2.244 

•'•'-44 
.244 

•244 
•a45 

2.245 

.245 

.245 

.246 

246 

2.246 
246 

247 
247 
a47 

2.247 
.248 
.248 
.248 
.248 

2.249 
.249 
.249 
.249 
.250 

2.250 
.250 
.250 
.250 
.251 

2.251 

■25" 
.251 

.252 
.252 

2.252 
.252 

-253 
.253 
.253 

a-253 
.254 
.254 
.254 
.254 


2a93 
4697 
7101 
9506 
191 2 

4318 
6725 
9132 
i54> 
3949 

6359 
8769 
1180 

359« 
6003 

8416 
0829 
3243 
5658 
8073 

0489 
2906 

53^3 
774« 
0159 

2578 
4998 

74'9 
9840 
2262 

4684 

7107 

953« 
1955 

4380 

6806 
9232 
1659 
4087 
6515 

8944 

1374 
3804 
6235 
8666 


2.255  '°99 


38 


39.90 
39.91 
39.92 

39-93 
39-94 

39-95 
39.96 

39-97 
39-98 
39-99 

40.00 
40.01 
40.02 
40.03 
40.05 

40.06 
40.07 
40.08 
40.09 
40.10 

40.11 

40.12 
40.13 
40.14 
40.15 

40.16 

40.17 
40.18 
40.19 

40.21 

40.22 
40.23 
40.24 
40.25 
40.26 

40.a7 
40.28 
40.29 
40.30 
40.31 

40.32 
40.34 
40.35 
40.36 
4    37 

40.38 
40.39 
40.40 
40.41 
40.42 

4°-43 
40.44 
40.46 
40.47 
40.48 

40.49 
40.50 
40.51 
40.52 
40.53 

40.54 
5U3 


110= 


log  M.  Wff.  1", 


2.255 
.255 
■a55 
-a55 
.256 

2.256 
256 
256 

257 
257 

2.257 

-257 
.258 
.258 
.258 

2.258 
.259 
-259 
■259 
.259 

2.259 
.260 
.260 
.260 
.260 

2.261 
.261 
.261 
.261 
.262 

2.262 
.262 
.262 
.263 
.263 

2.263 
.263 
.264 
.264 
.264 

2.264 
.265 
.265 
.265 
.265 

2.266 
.266 
.266 
.266 

.267 

i.267 
.267 
.267 
.268 
.268 

2.268 
.268 
.269 
.269 
.269 

2.269 


1099 

353a 
59'' 5 
8399 
0834 

3270 
5706 
8143 
0580 
3019 

5458 
7897 
0337 
2778 
5220 

7662 
0105 
2548 
4992 
7437 
9883 
2329 
4776 
7223 
9671 

aiao 

4570 
7020 

947' 
1922 

4374 
6X27 
9281 

«735 
4190 

6645 
9102 

»559 
4016 

6474 

8933 

«393 

3853 
6314 

8776 
1238 

370« 
6165 
8629 
1094 

3560 
6026 
8493 
0961 
3430 

5899 
8369 
0839 

33'° 
5782 

8255 


40.54 
40.55 
40.56 
40.58 
40.59 

40.60 
40.61 
40.62 
40.63 
40.64 

40.65 
40.66 
40.68 
40.69 
40.70 

40.71 
40.72 
40.73 
40.74 
40.75 

40.76 
40.78 
40.79 
40.80 
40.81 

40.82 
40.83 
40.84 
40.85 
40.86 

40.88 
40.89 
40.90 
40.91 
40.92 

40.93 
40.94 
40.95 
40.96 
40.98 

40.99 
41.00 
41.01 
41.02 
41.03 

41.0A 
41.06 
41.07 
41.08 
41.09 

41.10 
41.11 
41.12 
41.13 
41.15 

41.16 
41.17 
41.18 
41.19 
41.20 

41.21 


111= 


logM.         I    Diff.  1". 


8255 
0728 
3202 
5676 
8152 

0628 
3104 

5582 
8060 
0538 

3018 
5498 

7979 
0460 
2942 

54a5 
7909 

0393 
2878 

5364 

7850 

0337 
2825 

53«3 
7802 

0292 

2783 
5a74 
7766 
0258 

2.277  a75a 
277   5246 

277  7740 

278  0236 
278  2732 

2.278  5229 
.278  7726 
.279  0224 
.279  2723 
.279  5223 

2-279 
.280 
.280 
.280 
.280 

2.281 
.281 
.281 
.281 
.282 

2.282 
.2X2 
.282 
.283 
.283 


.269 

.270 
.270 
.270 
.270 

2.271 
.271 

.271 
.271 
.272 

2.272 
.272 
.272 
.273 
.273 

2.273 

.273 

-274 
.274 
.274 

2.274 
-a75 
-275 
.275 
.275 

2.276 
276 
276 
276 
277 


2.283 
.283 
.2X4 
.284 
.284 

2.284  7878 


7723 
0224 
2726 
5228 

773' 

0235 
2740 

5245 

775' 
0258 

2765 

5273 
7782 
0291 
2801 

5312 

7X24 
0336 
2849 
5363 


41.21 
41.23 
41.24 
41.25 
41.26 

41.27 
41.28 
41.29 
41.30 
41.32 

4'-33 
41.34 

4'-35 
41.36 
41.38 

41.39 
45.40 
41.41 
41.42 

4'-4: 

4 '-44 
41.46 

41.47 
41.48 
41.49 

41.50 
41.51 
4'-53 
4'-54 
4'-55 

41.56 

4'-57 
41. 58 
41.60 
41.61 

41.62 
41.63 
41.64 
41.65 
41.67 

41.68 
41.69 
41.70 
41.71 
41.72 

4'-74 

4'-75 
41.76 

4'-77 
41.78 

41. So 
4 1 .  X I 
41.82 
41.83 
41.84 

41.85 
41.87 
41.88 
41.89 
41.90 

41.91 


TABLE  VI. 

For  finding  the  True  Anom  ily  or  tlie  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


If 


V. 

112 

0 

# 

113 

0 

114° 

115° 

log  M. 

Dinr.  1". 

ItiK 

M. 

Diff.  1". 

log 

M. 

Dim  1". 

log  M. 

Dim  1". 

44,18 

0' 

1.284   7878 

41.91 

2.300 

0067 

42.64 

2-3«5 

4927 

43.40 

2-33' 

2564 

1 

.285 

0393 

41.93 

.300 

2626 

42.65 

•3«5 

753' 

43-41 

-33' 

5216 

44.20 

2 

.285 

2909 

41.94 

.300 

5186 

42.67 

.3.6 

0136 

43.42 

.331 

7868 

44.21 

3 

.285 

5415 

41.95 

.300 

7746 

42.68 

.316 

2742 

43-44 

-332 

0521 

44-2  2 

4 

.285 

7943 

41.96 

.301 

0307 

42.69 

.316 

5348 

43-45 

•332 

3'75 

44- -4 

5 

1.286 

0461 

41.97 

2.301 

2869 

42.70 

2.316 

7956 

43.46 

2-:,32 

^Jr 

44-2^- 

G 

.286 

2979 

41.99 

.301 

543« 

42.72 

-3'7 

0564 

43-47 

•332 

8485 

44- ^f- 

7 

.286 

5499 

42.00 

.301 

7995 

•i-'-73 

-3'7 

3173 

43-49 

•333 

1141 

44. 2  s 

H 

.286 

8019 

42.01 

.302 

0559 

42.74 

-3'7 

5782 

43-50 

-333 

3799 

44.29 

0 

.287 

0540 

42.02 

.302 

3123 

42-75 

-3>7 

8393 

43-5' 

-333 

6456 

44-3' 

10 

1.287 

3062 

42.03 

2.302 

5689 

42.76 

2.318 

1004 

43-53 

2-333 

9115 

44.32 

11 

.2S7 

5584 

42.04 

.302 

S255 

42.78 

.318 

3616 
6229 

43-54 

-334 

'775 

44-3  3 

12 

.287 

8107 

42.06 

.303 

0821 

42.79 

.318 

43-55 

•334 

4435 

44-3+ 

13 

.288 

0631 

42.07 

•303 

3390 

42.80 

.318 

8842 

43-56 

-334 

7096 

44-3<' 

14 

.288 

3'55 

42.08 

•303 

595S 

42.81 

-3'9 

1456 

43.58 

-334 

9 '58 

44-37 

15 

1.288 

5680 

42.09 

2.303 

8528 

42.83 

2.319 

4072 
6687 

43-59 

2-335 

2421 

44  39 

!     16 

.288 

8206 

42.10 

.3C4 

1098 

42.84 

.319 

43.60 

-335 

5084 

44.40 

1      IT 

.289 

0733 

42.12 

.304 

3668 

42.85 

.319 

9304 

43-62 

-335 

7749 

44-4' 

1      18 

.289 

3260 

42.13 

•3°4 

6240 

42.86 

.320 

1921 

43.63 

.336 

0414 

44-43 

I      11) 

.289 

5788 

42.14 

•3=4 

8i<;2 

42.88 

.320 

454° 

43.64 

-336 

3080 

44-44 

i     20 

t.289 

8117 
0847 

42.15 

2.305 

1385 

42.89 

2.320 

7'59 

43.66 

2.336 

5747 

44-45 

1     21 

.290 

42.16 

.305 

3959 

42.90 

.320 

9778 

43.67 

.336  8414 

44-47 

i     22 

.290 

3377 

42.18 

.305 

6533 

42.91 

.321 

2399 

43.68 

•337 

1.083 

4448 

23 

.290 

5908 

42.19 

.305 

9109 

42.93 

.321 

5020 

43.69 

•337 

3752 
6422 

44-49 

24 

.290 

8440 

42.20 

.306 

1685 

42.94 

.321 

7642 

43.70 

•337 

44-5' 

25 

:.29i 

0972 

42.21 

2.306  4261 

42.95 

2.322 

0265 

43.72 

2-337 

9093 

44.52 

26 

.291 

3505 
6039 

42.22 

.306 

6839 

42.96 

.322 

2889 

43-73 

-338 

1765 

44-53 

27 

.291 

42.24 

.306 

9417 

42.98 

.322 

55»3 

43-75 

.338 

4437 

44-55 

i     28 

.291 

8574 

42.25 

.307 

1996 

42.99 

.322 

8139 

43-76 

-338 

7111 

44.56 

29 

.292 

1 109 

42.26 

.307 

4576 

43.00 

•323 

0765 

43-77 

-338 

9785 

44.58 

1     30 

1.292 

6182 

42.27 

2.307 

7"7 

43.02 

2-323 

339' 

43-79 

2-339 

2460 

44-59 

;  31 

.292 

42.29 

.307 

9738 

43°3 

-323 

6019 

43.80 

-339 

5'35 

44.60 

1     32 

.292 

8719 

4230 

.308 

2320 

43-04 

•323 

8647 

43-81 

•339 

7812 

44,62 

i     33 

.293 

1258 

42.31 

.308 

4903 
7486 

43.05 

.324 

1277 

43-83 

.340  0490 

44.63 

'     34 

.293 

3797 

42.32 

.308 

43-07 

-324 

3907 

43.84 

.340 

3168 

44.64 

35 

1.293 

6336 

42.33 

2.309 

0071 

43.08 

2.324 

6537 

43.85 

2.340 

5847 

44.66 

36 

.293 

8877 

42.35 

.309 

2656 

43.09 

-324 

9169 

^Hl 

.340 

8527 

44.67 

!     37 

.294 

1418 

42.36 

.309 

5242 

43.10 

-325 

1801 

43.88 

-34' 

'^27 

44.69 

38 

.294 

3960 

42.37 

.309 

7828 

43.12 

.325 

4434 

43-89 

•3'r' 

3^89 

44.70 

39 

.294 

6503 

42.38 

.310 

0416 

43-'3 

•325 

7068 

43.91 

.341 

6571 

44-71 

40 

1.294 

9046 

42.40 

2.310 

3°34 

43.14 

2-325 

9703 

43.92 

2-34' 

9255 

44-73 

41 

.295 

1590 

42.41 

.310 

5593 

43-»5 

.326 

2339 

43-93 

•342 

1939 

44-74    . 

42 

.295 

HP 

42.42 

.310 

8182 

43-'7 

.326 

4975 

43-94 

.342 

4623 

44-75 

43 

.295 

6680 

42.43 

•3«i 

0773 

43.18 

.326 

7612 

43.96 

-342 

7309 

44-77     I 

44 

.295 

9227 

42.44 

.311 

3364 

43.19 

-327 

0250 

43-97 

-342 

9995 

44-78     i 

45 

2,296 

1774 

42.46 

2.311 

5956 

43.21 

2-327 

2889 

43.98 

2-343 

2683 

44.80 

;     46 

.296 

4321 

42-47 

.311 

8549 

43.22 

-327 

55J! 

44.00 

-343 

PV 

44.81 

1     *t7 

.296  6870 

42.48 

.312 

1142 

43-23 

.327 

8168 

44.01 

•343 

8060 

44.82 

48 

.296 

9419 

42.49 

.312 

3736 
6331 

43-24 
43.26 

.328 

0809 

44.0. 

•344 

0750 

44-84 

49 

.297 

1969 

42.51 

.312 

.328 

345' 

44.04 

•344 

3440 

44.85 

50 

2.297 

4510 

42.52 

2.312 

8927 

43-27 

2.328 

6094 

44.05 

2.344 

6132 

++■^1' 

51 

.297 

7071 

4*-S3 

•313 

1524 

43-28 

-328  8737 

44.06 

-344 

8824 

44. 8  S 

52 

.297 

9623 

42.54 

•3>3 

4121 

43.29 

•329 

1382 

44.08 

-345 

1517 

44.S9 

53 

.298 

2176 

42.55 

•3»3 

6719 

43-3» 

-329 

4027 

44.09 

-345 

4211 

44.91 

54 

.298 

4730 

4*'S7 

•3«3 

9318 

43-32 

■329 

6672 

44.10 

•345 

6906 

44.9a 

55 

2.298 

7284 

42.58 

2.314 

1917 

43-33 

2-329 

9319 

44.12 

2-345 

9601 

44-93 

56 

.298  9839 

42.59 

.314  4518 

43-35 

-330 

1967 

44.13 

•346 

2298 

44-93 

57 

.299 

=  395 

42.60 

.314 

7119 

43-36 

.330 

4615 

44.14 
44.16 

.346 

4995 

44.96 

58 

.299 

4952 

42.61 

.314 

9721 

43-37 

.330 

7264 

•346  7693 

44-97 

59 

.299 

7509 

42.63 

•3«S 

2323 

43-38 

.330 

9914 

44.17 

•347 

0392 

44-99 

60 

2.300 

0067 

42.64 

2.315 

4927 

43.40 

2-33' 

2564 

44.18 

2-347 

3092 

45.00 

694 


TABLE  VI. 

Vor  finding  tJie  True  Anoniiily  or  the  Tinu'  from  the  Perihelion  in  ii  Parabolic  Orbit. 


l\ 


1 
'i 

a 

4 

5 
6 

7 
H 
0 

10 
11 
12 
13 
14 

15 
lA 
17 
18 
lU 

20 
21 
22 
23 
24 

25 
20 

27 
28 
29 

30 
31 
32 
33 
34 

35 
30 
37 

38 
3» 

40 
41 
42 
43 
44 

45 
40 
47 

48 
4» 

50 
51 
52 
53 
54 

55 
50 

57 
58 
59 

GO 


116^ 


l"K  M. 


2-347 
347 
347 

34« 
348 
34X 
349 
34'; 
349 

3^0 

3'io 

350 
350 
35' 

351 
351 
35« 

352 
35* 

352 
352 
353 
353 
353 

354 
354 
354 
354 
355 

355 
355 
355 
356 
356 

356 

357 
357 
357 
357 


3091 
579* 
»494 
1 196 
3*199 

6603 
9308 
Z014 
4720 
7428 

0136 
2845 

5554 
8265 

C977 

3689 
6402 
91 16 
1831 

4547 

7263 
9981 
2699 
5418 
8138 

0859 
3581 
6303 

9027 

1 75 1 

4476 
7202 
9928 
2656 
53«S 
8114 
0844 

3575 
6307 
9040 


Diff.  1" 


358  1773 
358  4508 

358  7243 

358  9979 

359  2716 


359 
359 
360 
360 
360 

360 
36, 
361 
361 
362 

362 
362 
362 
363 
363 
363 


5454 
8193 
0933 

3673 
6415 

9157 
1900 

4644 
7389 

0134 

2881 
5628 
8376 
1 1 26 
3876 

6626 


45.00 
45.02 
45.03 
45.04 
45.C6 

45-07 
45.09 

45.10 
45.11 
45-«3 

45-»4 
45.16 

45-«7 
4518 
45.20 

45-21 
45-23 
45-24 
45-25 
45-27 

45.28 
45.30 
45-3> 
45-33 
45-34 

45-35 
45-37 
45.3« 
45-40 
45-41 

45-42 
45-44 
45-45 
45-47 
45.48 

45.50 
45-5' 
45-52 
45-54 
45-55 

45-57 

45-58 
45.60 
45.61 
45.62 

45.64 
45-65 
45-67 
45.68 
45.70 

45-7» 
45-72 
45-74 
V5-75 
45-77 

45.78 
45.80 
45.81 
45.82 
45.84 

45.86 


117^ 


I<>K  M. 

363    6626 

363  9378 

364  2131 

364  48X5 
364  7639 


365 
365 
365 
365 
366 

366 
366 
366 

367 
367 

367 
368 
368 
368 
368 

369 

369 
369 

370 
370 

370 

370 
371 
371 
371 

371  9559 

372  2337 
372  51 16 

372  7896 

373  0677 

373 
373 
373 
374 
374 


°394 

3'5o 
5907 
8665 

1423 

4183 

6944 
9705 
2467 
5230 

7994 

0759 
3525 
6291 
9059 

1827 
4596 

7367 
0138 
2909 

5682 
8456 
1230 
4006 
6782 


3459 
6241 
9024 
1809 
4594 

7380 
0167 
2955 
5744 
8533 

1324 
4115 
6908 
9701 
2495 


374 
375 
375 
373 
375 

376 
376 
376 
376 

377 

377  5290 

377  8086 

378  0883 
378  368? 
378  6479 

378  9279 

379  2079 
379  4881 

379  7683 

380  0486 

380  3290 


Dlff.  1". 

45-86 
45.87 
45.88 
45.90 
45-9« 

45-93 
45-94 
45.96 

45-97 
45-99 

46.00 
46.01 
46.03 
46.04 
46.06 

46.07 
46.09 
46.10 
46.12 
46.13 

46-15 
46.16 
46.18 
46.19 
46.21 

46.22 
46.24 
46.25 
46.26 
46.28 

46.29 
46.31 
46.32 
46.34 
46.35 

46.37 
46.38 
46.40 
46.41 
46-43 
46.44 
46.46 
46.47 
46.49 
46.50 

46.51 
46.53 

46-55 
46.56 
46.58 

46.59 
46.60 
46.62 
46.64 
46.65 

46.67 
46.68 
46.70 
46.71 
46.73 

46.74 


118= 


loK  M. 


Dlff.  I". 


2.380  3290 
.380  6095 
.380  8901 
.381  1708 
.381    4515 


2.381 
.382 
.3X2 
.3S2 
.382 

2-383 
•383 
.38, 
.383 
-384 

2.384 
-384 
•385 
•385 
.385 

2.385 
.386 
-386 
.386 
.387 

2.387 

-387 
.387 
.388 
.388 

2.388 

-389 
-389 
.389 
.389 

2.390 
.390 
-390 

-39- 
.391 

2.391 
.392 
.392 
.392 
.392 

2-393 
-393 
-393 
-393 
•394 

2-394 
-394 
-395 
•395 
•395 

2-395 
.396 
.396 
.396 
-397 


7324 
o'33 
2944 

575  5 
8567 

1380 

4 '94 
7009 
9825 
2642 

5460 
8278 
1098 
3918 
6739 

9562 

2385 
5209 

8034 
0860 

3687 
651.1 

9343 
2173 
5003 

7835 
0607 
3500 

6335 
9170 

2006 
4843 
7681 
0519 
3359 
6200 
9042 
1884 
4728 
7572 

0417 
3264 
61 1 1 
8959 

1808 

4658 
7509 
0361 

0067 

8922 

1778 
4634 

7492 
0350 


2-397  3210 


46-74 
46.76 

46.77 
46.79 
46.80 

46.82 
46.83 
46.85 
46. 86 
46.88 

46.89 
46.91 
46,92 
46.94 
46.95 

46.97 
46.98 
46.99 

47.01 
/I  •'.03 

..-05 
47.06 
47-08 
47.09 
47.11 

47.12 
47.14 
47- '5 
47-'7 
47.18 

47-20 
47.21 

47-23 
47.24 
47.26 

47.28 
47.29 

47-3' 
47-32 
47-34 

47-35 

47-37 
47.38 
47.40 
47.41 

47-43 
47-45 
47.46 
47.48 
47-49 

47-5' 
47.52 

47-54 
47-55 
47-57 

47-59 
47.60 
47.62 

47-63 
47.65 

47.66 


119^ 


logM. 


i97 

397 
397 
398 

398 

398 
399 
399 
399 
399 

400 
400 
400 
401 
401 


3210 

6070 
8931 

1794 
4657 

7521 
0386 

325^ 
6119 
8987 

1856 

4725 

759'! 
046S 

3340 

401  6214 

401  90S 8 

402  1964 
402  4840 

402  7718 

403  0596 

403  3475 
403  6356 

403  9237 

404  2 1 1 9 

404  5002 

404  7886 

405  0771 
405  3657 
405  6544 


405 
406 
406 
406 

407 

407 
407 
407 
408 
408 


9432 
2321 
521 1 
8102 
0993 

3886 
6780 
9674 
2570 
5467 


408  8364 

409  1263 
409  4162 
409  7063 

409  9964 

410  2866 
410  5770 

410  8674 

411  1579 

411  4486 

4"  7393 

412  0301 
412  3210 
412  6120 

412  9031 

413  1944 
413  4857 

413  7771 

414  0686 
414  3602 

414  6519 


Dlff.  1". 

47.66 
47.68 
47.70 
47-7' 
47-73 

47-74 
4776 
47-77 
47-79 
47.81 

47-82 
47.84 
47.85 
47-87 
47-89 

47.90 
47.92 
47-93 
47-95 
47-97 

47.98 
48.00 
48.01 
48.03 
48.04 

48.06 
48.08 
48.09 
48.11 
48.12 

48.14 
48.16 
48.17 
48.19 
48.20 

48.22 
48.24 
48.25 
48.27 
48.28 

48.30 

48-32 
48.33 

48.35 

48-37 

48.38 
48.40 
48.41 
48-43 
48-45 

48.46 
48.48 
48.49 
48.51 
48-53 

48.54 
48.56 
48.58 
48.59 
48.61 

48.62 


6V5 


TABLE  VI. 

For  finding  tlie  True  Anomaly  or  the  Time  from  tlie  Perihelion  in  a  Parabolic  Orbit. 


V. 


o 
1 
•z 

3 
4 

5 
0 

7 
8 
O 

10 
11 
1'^ 
13 
14 

15 
16 
17 

18 
lU 

20 
'Zl 
TZ 
23 
24 

23 
2A 
27 

28 
2U 

30 
31 
32 
33 
34 


30 
37 

38 
39 

40 
41 
42 
43 
44 

45 
40 

47 
48 
49 

30 
51 
32 
53 
54 

55 
36 
37 

58 
59 

00 


120^ 


log  M.  Dlir.  1". 


4'4 
4'4 
4'5 
4' 5 
4>S 
416 
416 


6519 

941 7 

5276 
8197 

1119 

4042 


416  6965 

416  9890 

417  2816 

417   5743 

417  8671 

418  1600 
418  4529 

418  7460 

419  0392 
419  3325 
419  6258 

419  9193 

420  2129 

420  5066 
^zo   8003 

421  0942 


48.62 
48.64 
48.66 
4X.67 
48.69 

48.71 
48.72 
48.74 
48.76 

48.77 

48.79 
48.81 
48.82 
48.84 
48.85 

48.87 
48.89 
48.90 
48.92 
48.94 

48.95 
48.97 
48.99 


421  3882  .  49.00 

421  6822  I  49.02 

421  9764  i  49.03 

422  2707  I  49.05 
422  5650  \  49.07 

422  8595  I  49.09 

423  1541  j  49.10 

423  4488  I  49.12 


423  7435 

424  0384 

4^4  3  334 
424  6284 

424  9236 

425  21X9 

425  S'42 

425  8097 

426  1053 

426  4010 
426  6967 

426  9926 

427  2886 
427  5847 


49.14 
49.15 
49.17 
49.19 

49.20 
49.22 
49.24 
49.25 
49.27 

49.29 
49.30 
49-3* 
49-34 
49-35 


427  8808  I  49.37 

428  1771  i  49.39 


.28 

.28 


4735  I  49-40 
7700  i  49.42 


429  0665  I  49.44 

429  3632  :  49.46 

429  6600  i  49.47 

429  9569  I  49.49 

430  2539  [  49.51 
430  5510  !  49.52 

430  8482  I  49.54 

43«  «455  49-56 


431  442!- 


49-57 


43>  7403  !  49-59 
43*  0379  I  49-6i 

43*  3356  j  49-62 


121' 


lot?  M.    Via.  1", 


2.432 

431 
432 

433 
433 

433 
434 
434 
434 
435 


\l 


3356 
6334 
93'3 
2293 

5274 

8257 
1240 
4224 
7209 
0195 

182 

I7« 

9160 
2150 
5141 

8134 
1127 
4122 
7117 
01 14 

3111 

6110 
9109 


435 
435 
43  5 
436 
436 

436 
437 
437 
437 
438 

438 

43« 
43« 

439  2IIO 

439  5««2 
439  81J4 


440 


1118 


440  4123 
440  7129 


441  01 


36 


441  3143 
441  6152 

441  9162 

442  2173 
442  5185 

442  8199 

443  1 21 3 
443  4228 

443  7244 

444  0261 

444  3280 
444  6299 

444  9320 

445  2341 
445  5364 

445  8387 

446  1412 

446  4437 

446  7464 

447  0492 

447  3521 
447  6551 

447  9582 

448  2614 
448  5647 

448  8681 

449  '7«6 
449  47  53 
4-V9  7790 


45' 


0828 


450  3 


868 


49.62 
49.64 
49.66 
49.68 
49.69 

49.71 
4973 
49-74 
49.76 
49.78 

49.80 
49.81 
49.83 
49.S5 
49.86 

49.88 
49.90 
49.92 
49-93 
49-95 

49-97 
49.98 

50.00 
50.02 
50.04 

50.05 
50.07 
50.09 
50.1 1 
50.12 

50.14 
50.16 
50.18 
50.19 
50.21 

50.23 
50.24 
50.26 
50.28 
50.30 

50.31 

50-33 

50-35 
50.37 
50.38 

50.40 
50.42 
50.44 

50-45 
50.47 

50.49 
50-5 « 
50-53 
50-54 
50.56 

,  -58 
50.60 
50.61 
50.63 
50.65 

50.67 


122' 


lo«  M.        I 

450  3868 
450  6908 

450  9950 

451  2992 
451  6036  ' 

451  9081 

452  2127 

452  5>74 

452  8222  . 

453  J27«  I 


453 
453 
454 
454 
454 

454 
455 
455 
455 
456 

456 
456 
457 


4321  I 

7372  ■ 
0424 
3477  . 
6532  : 

9587  i 
2644 

5701 
8760 
1820  { 

4881  I 

7943 
1006  i 


45X  3 


l68 


Dim  1". 

50.67 
50.68 
50.70 
50.72 
50.74 

50-75 
50-77 
50.79 

50.81 

50.83 

50-84 
50.86 
50.88 
50.90 
50.92 

50-93 
50.95 

50-97 
50.99 

51.00 


457  4070  ,  5 

457  7»35  I  5 

458  0201  '  5 


5 


458  6337  I  5 

458  9406  j  5 

459  2477  I  5 

459  5548  i  5 

459  8621  ,  5 

460  1695  5 
460  4770  j  5 

460  7846  I  5 

461  0923   5 
461  4001  I  5 

461  7080  i  5 

462  0161  ;  5 
462  3242  I  5 

462  6325  j  5 

462  9408  '  5 

463  2493  ;  5 

463  5579  '  5 

463  8666   5 

464  1754 

464  4843 

464  7933 

465  1024 
465  4116 

465  7210 

466  0305 
466  3400 
466  6497 

466  9595 

467  2694 

467  5794 

467  8895 

468  1997 
468  5101 

468  8205 


.04 
.06 
.08 
.09 

.11 
-'3 
-«5 
-17 
.18 


24 
26 
28 


33 
35 

37 


.40 

.42 


48 
49 
51 

53 
■55 


.60 
.62 
.64 

.66 
.68 

■70 
-7» 
•73 


51-75 


123° 


10(5  M.         I    Dlir.  1".   I 


470  3744 
470  6856 

470  9968 

471  3081 
471  6196 


468 
469 
469 
469 
470 


8205 
1311 
4418 
7526 
0634 


471 

472 
4.72 
•472 
473 

473 
473 
474 
474 
474 


93" 

2428 

5546 
8665 
1785 

4906 
8028 
1152 
4276 
7402 


475  0529 
475  3657 
475  6786 

475  99«6 

476  3047 

476  6180 

476  9313 

477  2448 
477  5584 

477  8721 

478  1859 
478  4998 

478  8138 

479  1280 
479  4422 

479  7566 

480  07 1 1 
400  3857 

480  7004 

481  0152 

481  3301 
481  6452 

481  9604 

482  2756 
482  5910 

482  9065 

483  2222 

483  5379 

483  8537 
•1-84  1697 

484  4858 
48.1.  8020 

485  1183 
485  4347 

485  75«3 

486  0679 
486  3847 

486  7016 

487  0186 
487  3357 
487  6529 


«-75 
'-77 
1.79 
I. Si 
1.82 

1.84 
1.86 
1.88 
1.90 
1.92 

1.94 
'-95 
'•97 
•-99 
2.01 

2.03 
2.05 
2.07 
2.09 
2.10 

2.12 
2.14 
2.16 
2.18 
2.20 

2.22 
2.23 

2.25 
2.27 
2.29 


2-33 
2-35 
2.37 
2.39 

2.40 

2.42 

2-44 
2.46 
2.48 

2.50 
2.52 
2.54 
2.56 
2.58 

2-59 
2.61 

2.63 
2.65 
2.67 

2.69 
2.71 

2.73 
2-75 
2.77 

2.78 
2.80 
2.82 
2.84 
2.86 


52.; 


590 


rabolic  Orbit. 


TABLE  VI. 

For  finding  tlif  True  Anomaly  or  tiie  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


123 

0 

> 

OB  M. 

DIfT.  1". 

5«-75 

8   8205 

9   »3>i 

5'-77 

9  44»8 

5  ••79 

9  75i6 

51. Xi 

0  0634   , 

51.82 

°  3744  1 

51.84 

0  6856 

51.86 

0  9968 

51.X8 

I    3081 

51.90 

I   6196   1 

51.92 

1   9311 

S>-94 

2  2428 

51-95 

2  5546 

51-97 

2  8665 

51.99 

3  1785 

52.01 

3  4906 

52.03 

3  8028 

52.05 

4  «i52 

52.07 

4  4276 

52.09 

4  740* 

52.10 

5  0529 

52.12 

5  3657 

52.14 

5  6786 

52.16 

5  99'6 

52.18 

6  3047 

52.20 

6  6180 

52.22 

6  9313 

52.23 

7  2448 

52.25 

7  5584 

52.27 

7  8721 

52.29 

r8   1859 

52-3' 

8  4998 

52-33 

8  8138 

52-35 

9   1280 

52-37 

r9  4422 

52-39 

r<)   7566 

52.40 

0  0711 

52.42 

,0  3857 

52-44 

!o  7004 

52.46 

!i  0152 

52.48 

!i   3301 
!i   6452 

52.50 

52.52 

ii   9604 

52-54 

iz  2756 

52.56 

fz  5910 

52.58 

?2    9065 

52-59 

!3  2222 

52.61 

|3  5379 

52-63 

«3  8537 

52-65 

<4  1697 

52.67     , 

^  4858 

52.69 

?.1.  8020 

52-71     ' 

^5   "83 

52-73     . 

'5  4347 

52-75     j 

is  7513 

52-77     1 

S6  0679 

52.78     1 

i6   3847 

52.X0    ! 

i<6  7016 

1   52-82    1 

S7  oi86 

i  52-84  ! 

^7  3357 

52.86   I 

87  6529 

52.88 

V. 


O' 

1 

:i 
4 

5 
0 

7 

8 
9 

10 
11 
Vi 
13 
14 

15 
10 
17 

18 
10 

20 
21 
22 
2:1 
24 

25 
20 
27 

28 
20 

30 
31 
32 
33 
34 

35 
30 
37 

38 
30 

40 
41 
42 
43 
44 

45 
40 

47 
48 
40 

50 
51 
52 
53 
54 

55 
50 
57 
58 
50 

60 


124^ 


IngM. 


487 

487 
488 
488 
488 

489 
489 
489 
490 
490 

490 
491 
491 
491 
492 

492 
492 
493 
493 
493 

494 
494 
494 
494 
495 

495 
495 
496 
496 
496 

497 
497 
497 
498 
498 

498 
499 
499 
499 
500 

500 
500 
501 
501 

501 

502 
502 
502 
503 
503 

503 
503 

504 
504 
504 

5°S 
5°5 

506 
506 


6529 

9702 
2877 
6053 
9230  i 

2408  I 
5587 
8767   ' 
«949 
5132   ^ 

8315  ; 
1 500  i 
4686  '. 

7874 
1063  I 

4252  I 

7443  ; 
0635  j 
3828  I 
7023  ' 

0218 

34«5 
6613 
9812 

3012 

6213 
9416 
2619 

5824 
9030 

2238 

5446 
8656 
1867 
5079 

8292 
1506 
4721 

7938 
1156 

4375 
7595  , 
0817 

4°39  ' 
7263  ! 

0488  I 

37«4  ' 
6942 

0170  i 

3400  j 

6631 
9863 
3096 

633' 
9567 

2804 
6042   i 
9282   I 
2522 
5763 


DIff.  1". 


2.88 
2.90 
2.92 
2.94 
2. 90 

2.98 

3.00 
3.02 
3.03 
3.05 

3.07 
3.09 
3.11 
3-'3 
3-J5 

3->7 
3.19 

3.21 

323 

325 

3-27 
3.29 

3-3' 
3-33 
3-35 

3 
3 


506  9006 


37 
39 
3-41 
3.42 

3-44 

3-46 
3.48 
3.50 

3-52 
3-54 

3.56 
3.58 
3.60 
3.62 
3.64 

3.66 
3.68 

3.70 
3.72 
3-74 

3.76 
3.78 
3.80 
3.82 
3.84 

3.86 
3.88 

3-9° 
3.92 

3-94 

3.96 
3.98 
4.00 
4.02 
4.04 

54.06 


125^ 


loK  M. 


2.506 

.507 
.507 
.507 
.508 

2.508 
508 
"509 
509 
509 


2-5 

5 
5 
5 
5 

2-5 

5 
5 
5 
5 

2-5 

5 
5 
5 
5 

2-5 

5 
5 
5 
5 

2-5 
5 
5 
5 
5 

2-5 

5 
5 
5 

5 


9006 
2251 
5496 
8742 
1990 

5239 
84S9 

•74> 
4993 
8247 

1502 

4758 
8016 
1274 
4534 

7795 

1057 

4321 

2  7586 

3  0852 

3  4««9 

3  7387 

4  0657 
4  3927 

4  7«99 

5  0473 
5  3747 

5  7023 

6  0300 
6  3578 

6  6857 

7  0138 

7  3420 
7  6703 

7  9987 

8  3273 
8  6559 

8  9847 

9  3'37 
9  6427 


2.519  9719 

520  3012 
520  6306 

520  9601 

521  2898 

2.521  6196 
.521  9495 
.522  2795 
.522  6097 
.522  9400 

2.523  2704  j 
.523  6009  I 
.523  9316  i 
.524   2624  ■: 

-524  5933  ' 

2.524  9243 

•525  2555 
.525  5867 
.525  9181 
•5*6  *497 

2.516  5813  I  55.29 


Dlflr.  1". 


4.06 
4.08 
4.10 
4.12 
4.14 

4.16 
4.18 
4.20 
4.22 
4.24 

4.26 
4.28 
4.30 
4.32 
4-34 
4.36 
4.38 
4.40 
4.42 
4-44 
4.46 

4.48 
4.50 

f52 

4-54 
4.56 
4.58 
4.60 
4.63 
4-65 

4.67 
4.69 
4-71 
4-73 

4-75 

4-77 

4-79 
4.81 
4.83 
4.85 

4-87 
4.89 

4.91 
493 
4-95 

4-97 
4-99 

5.02 
5.04 
5.06 

5.08 
5.10 

5-12 

5.14 
5.16 

5.18 
5.20 
5.22 
5.24 
5.26 


126^ 


li>K  M. 


DilT.  1". 


526    5813 

526  9131 

527  2450 

527   577« 

527  9092 

528  2415 

528   5739 

528  9065 

529  2391 
529  5719 

529  9048 

530  2379 

530  5710 

5  3°  9043 
53>  2378 

531  5713 
53'  9050 

532  2388 

532  5727 
532  9068 


533 
533 
533 
534 
534 

534 
535 
535 

536 

536 
536 

537 
537 
537 

53! 

"! 
538 

539 

539 


2410 

5753 
9097 

2443 

579° 

9138 

2487 

5838 
9190 

"-5+3 

5898 

9254 
261 1 

5970 
9329 

2690 
6052 
9416 
2781 
6147 


54' 
54' 
542 
542 
542 

543 
5-^3 
544 
544 
544 


539  95'4 

540  2883 
540  6253  i 

540  9625  ! 

54'  2997 


6371 

9746 
3123 
6500 
9880 

3260 
6641 
0024 
3409 
6794 


545  0181 
545  3569 

545  6959 

546  0350 
546  3742 

546  7135      56.57 


55-29 
5  5-3' 
55-33 
55-35 
55-37 

55-39 
55-41 
55-43 
55-45 
55-48 

55-5° 
55-52 
5  5-54 
55-56 
55-58 

55.60 
55-62 
55-64 
55-67 
55.69 

55-7' 
55-73 
55-75 
55-77 
55-79 
55.8, 
55-84 
55.86 
55-88 
55.90 

55.92 

55-94 
55.96 
55.98 
56.01 

56.03 
56.05 
56.07 
56.09 
56. II 

56-»3 
56.15 
56.18 
56.20 
56.22 

56.24 
56.26 
56.29 
56-31 
56-33 

56-35 
5637 
56-39 
56.42 

56-44 

56.46 
56.48 
56.50 
56.52 
56.55 


127 

0 

luK  M. 

Diir.  1". 

2  546  7'35 

56-57 

■547  0530 

56.59 

-547   3926 

56.61 

547  7323 

56.63 

5.!8  0722 

56.65 

2.548  4122 

56.68 

.548  7523 

56.70 

•549  0926 

56.72 

-549  4330 

5^'+ 

-549  7735 

56.76 

2.550  1141 

56.79 

-55°  4549 

56..S1 

-55°  7958 

56-83 

-55'    '369 

56.S5 

•55'   4781 

56-87 

2.551   8194 

56.90 

.552   1608 

56.92 

-552  5024 

56.94 
56.96 

.552  8441 

.553   '859 

56.98 

2.553   5279 

57-01 

-553  8700 

57^o3 

•554  2122 

57^os 

.554  5546 

57^07 

•554  897' 

57^io 

2.555  2398 
•555   5825 

57.12 

57.T4 
57.16 

.555  9254 

.556  2685 

57.18 

.556  61 16 

57.21 

2.556  9549 

57-23 

.557  2984 

57-25 

•557  6420 

57-27 

•557  9857 

57-29 

.558  3295 

57-32 

2.558  6735 

57-34 

•559  °«76 

57-36 

■559   36'8 

57-38 

•559  7062 

57.4' 

.560  0507 

57.43 

2.560  3953 

57-4S 

.560  7401 

57.47 

.561   0850 

57.50 

.561   4301 

57-52 

.561   7753 

57-54 

2.562   1206 

57.56 

.562  4660 

57-59 

.562  8116 

57-61 

.563    1574 

57-63 

.563   5032 

57-65 

2.563   8492 

57-68 

.564  1953 
.564  5416 

57-70 

57-72 

.564  8S80 

57-74 

.565   2345 

57-77 

2.565   5812 

57-79 

.565  9280 

57-81 

.566  2750 

57.8a 
57.86 

.566  6221 

.566  9693 

57.88 

2.567  3166 

57.90 

597 


TABLE  VI. 

For  fmiHnjt  the  Triio  Aiiomiily  (ir  llie  I'iiiio  rioiii  tlic  Purihelidii  in  a  Paraljolio  Orbit. 


1 

1 

128 

0 

•129 

0 

130 

\° 

131 

0 

V, 

2.^(17  5ifi6 

DifT.  1". 

57-90 

1<>K  M.         ! 

_  1 

2.5S8   4112 

Dlff.  1". 

1.>K  M. 

Dlff.  1". 

60.75 

loR  M. 

2.632   1622 

Diff,  1". 

59.30 

2.610   0188 

62.28     : 

0 

1    > 

.567  6641 

57-9  3 

.588    7670 

59-32 

.610   3834 

60.78 

.632  5360 

62.30 

I 

1 

1       2 

.?(>S  0117 

57-95 

.589    1230    ' 

59-35 

.610  '7481 

60.80 

.632  9099 

^*-33      , 

1 

2 

'i 

•56«   3595    , 

57-97 

.589   4792 

59-37 

.611    1110 
.611    4781 

60.83 

.633  2839 

62.35 

1 

3 

!    * 

.568  7074 

57-99 

.589   831:5 

59-39 

60.85 

.633  6581 

62.38 

1 

i     4 

'       5 

2.56,)  0554 

58.02 

2.590    1919 

59-4- 

2.6 1 1    8433 

60.88 

2.634  0325 

62.41 

1 

i     5 

» 

■5''9  403'' 

58.04 

.590   5485 

59-V4 

.612  2086 

60.90 

.634  4070 

62.43 

I 

0 

7 

,       7 

.S(>9  75 "J 

58.06 

.590   9052 

59-47 

.612   5741 

60.93 

.634  7817 

62.46 

1 

1       H 

.S70   1004 

58.09 

.591    2620 

59-49 

.612   9397 

60.95 

.635  1565 

62.48 

H 

H 

1       » 

.S70  449° 

58,11 

.591    6190 

59-5' 

.613    3055 

60.98 

-635   53«5 

62.51 

■ 

0 

10 

1-570  7977 

58.13 

2.591    9762    1 

59^54 
59.56 

2.613   6715 

61.00 

2635  9066 

62.54 

■ 

'     10 

11 

•57"    H65 

58.15 

-592   33  35 

.614  0376 

61.03 

.636  2819 

62.56 

H 

;      11 

12 

■57"   4955 

58.1S 

.592  6909 

59.58 

.614  4038 

61.05 

.636  6573 

62.59 

I 

12 

13 

.571   8447 

58.20 

•593   0485 

59.61 

.614  7702 

61.08 

.637  0329 

62.61 

H 

13 

It 

.572   1939 

58.22 

•593  4062  1 

59.63 

.615    1368 

61.10 

.637  4087 

62.64 

■ 

'     14 

15 

2-572   5434 

58.25 

2.593  7641 

59.66 

2.615    5035 

61.13 

2.637  7846 

62.67 

I 

1     15 

1» 

-572  «929 

58.27 

-594   •"! 

59.68 

.615   S703 

61.15 

.638   1607 

62.69 

■ 

10 

17 

-573   1426 

58.29 

-594  4803    ■ 
•594  83S6 

59.70 

.61(1   2373 

61.78 

.638  5369 

62.72 

H 

17 

18 

•573   59^4 

58-3* 

59-73 

.616   6045 

61.20 

.638  9133 

62.75 

■ 

18 

10 

-573  94*4 

58.34 

•595   '970 

59-75 

.616  9718 

61.23 

.639  2899 

62.77 

■ 

10 

20 

2-574  2925 

58.36 

2-595   5556 

59.78 

2-617   3392 

61.25 

2.639  6666 

62.80 

■ 

20 

21 

-574  6427 

58.38 

•595   9>43 

59.80 

.617   70(18 

61.28 

.640  0435 

62.82 

H 

21 

22 

-574  993« 

5S.41 

.596  2732 

59.82 

.618  0746 

61.30 

.640  4205 

62.85 

H 

22 

23 

•575   3436 
-575  6943 

58-43 

.596  6322 

59^85 

.618  4425 

6«-33 

.640  7977 

62.88 

H 

23 

24 

58.45 

.596  9914 

59-87 

.618   8105 

61.36 

.641    1750 

62.90 

■ 

24 

25 

2.576  04s  I 

58.48 

2-597   3507 

59.90 

2.619   '787 

61.38 

2.641   5525 

62.93 

1 

25 

20 

.576  3960 

58.50 

-597  7'02 

59.92 

.619   5471 

61.41 

.641   9302 

62.96 

^1 

2(» 

27 

.576  7471 

58.52 

.598  0698 

59-95 

.619  9156 

61.44 

.642  3080 

62.98 

H 

27 

28 

-5  77   09**  3 

58-55 

•598  4295 

59-97 

.620  2843 

61.46 

.642  6860 

63.01 

S 

28 

2» 

j 

-577  4496 

58-57 

.598   7894 

59-99 

.620  6531 

61.48 

.643  0641 

63.04 

H 

20 

30 

1.577   8oi  I 

58.59 

2.599   1494 

60.02 

2.621   0220 

61.51 

2.643  4424 

63.06 

1 

30 

!     31 

.578   1528 

58.62 

•599  5^96 

60.04 

.621    3911 

61.53 

.643  8209 

63.09 

^^ 

31 

32 

.578   5045 

58.64 

•599  8699 

60.07 

.621    7604 

61.56 

.644  1995 

63.1a 

^1 

32 

;     33 

.578   8564 

58.66 

.600  2304 

60.09 

.622   1298 

61.58 

.644  5783 

63.14 

^1 

33 

34 

•579  »o8S 

58.69 

.600  5910 

60.12 

.622  4994 

61.61 

.644  957a 

63.17 

H 

34 

35 

2-579   5607 

58-71 

2.600  9518 

60.14 

2.622   8691 

61.63 
61.66 

2.645  3363 

63.19 

H 

35 

30 

-579  9130 

58-73 

.601    3127 

60. 1 6 

.623   2390 

-645  7>55 

63. -".i 

^1 

30 

37 

.580  2655 

5^7^ 

.601   6738 

60.19 

.623   6091 

61.68 

.646  0949 

63.25 

H 

37 

38 

.580  6181 

58.78 

.602  0350 

60.21 

.623   9793 

61.71 

.646  4745 

63.27 

SB 

!    38 

39 

.580  9708 

58.80 

.602   3963 

60.24 

.624   3496 

61.74 

.646  8542 

63.30 

H 

30 

10 

2.5S1    3237 

58.83 

2.602  7578 

60.26 

2.624  7201 

61.76 

2.647  2341 

63.33 

H 

40 

41 

.581   6768 

58-85 

.603    1 195 

60.29 

.625   0907 

61.79 

.647  6142 

63-35 

^1 

41 

42 

.5 82  0299 

58.87 

.603  4813 

60.31 

.625   4615 

61.81 

.647  9944 

63.38 

^1 

42 

43 

.582   3832 

58.90 

.603   8432 

60.34 

.625   8325 

61.84 
61.86 

.648  3748 

63.4. 

^1 

43 

44 

.582   8267 

58.92 

.604  2053 

60.36 

.626   2036 

.648  7553 

63.44 

H 

44 

45 

2.583   0903 

58.94 

2.604  5675 

60.38 

2.626   5748 

61.89 

2.649  1360 

63.46 

H 

45 

1     4G 

.583  4440 

58.97 

.604  9299 

60.41 

.626  9462 

61.91 

.649  5168 

63.49 

^1 

40 

47 

-583   7979 

58-99 

.605   2924 

60.43 

.627   3178 

61.94 

.649  8978 

63.52 

^H 

47 

48 

.584   1519 

59.01 

.605   6551 

60.46 

.627   6895 

6i.>.- 

.650  2790 

63-54 

^H 

48 

40 

.584  5061 

59.04 

.606  0179 

60.48 

.628  0614 

61.99 

.650  6603 

63-57 

H 

40 

50 

2.584  8604 

59.06 

2.606  3S09 

60.51 

2.628  433J. 
.628   8056 

62.02 

2.651  0418 

63.60 

H 

50 

51 

.585   2148 

^9.09 

.606  7440 

60.53 
60.56 

62.04 

.651  4235 
.651   8053 

63.62 

^H 

51 

i     52 

.585   5694 

'59.11 

.607   1073 

.629    1780 

62.07 

63.65     : 

^H 

52 

53 

.585   9241 

S9-'3 

.607  4707 

60.58 

.629   5505 

62.09 

.652  1873 

63. 6X 

^H 

53 

54 

.586  2790 

59.16 

.607  8343 

60.61 

.629  9231 

62.12 

.652  5695 

63.70 

^H 

54 

55 

2.586  6340 

59.18 

2.608    1980 

60.63 

2.630  2959 

62.15 

2.652  9518 

63.73 

^M 

55 

50 

.586  9891 

59.20 

.608   5618 

60.66 

.630  668g 

62.17 

•653  334* 

63.76 

HI 

50 

57 

-587   3444 

59.23 

.608  9258 

60.68 

.631   0420 

62.20 

.653  7168 

63.79 

■■ 

1    57 

58 

.587  6999 

59-^5 

.609   2901 

60.70 

.631   4152 

62.22 

.654  0096 

63.>!i 

^H 

58 

50 

.588  05S5 

59.27 

.609  6544 

60.73 

.631   7887. 

62.25 

.654  4826 

63-84 

^M 

50 
00 

60 

2.588  4112 

59.30 

2.610  0188 

60.75 

2.632  1622 

62.28 

1 

2.654  8657 

63.87 

■ 

598 


iliolic  Orbit. 


TABLE  VI. 

For  fiiidinp  the  Tnie  Anoniiily  or  tlif  Tiiiu'  from  the  Perilielioii  in  a  I'uriiholic  ()ri)il. 


131 

0 

M. 

Diff.  1". 

1621 

62.28 

5360 

62.30 

i)0<)l) 

62.33   i 

2S39 

62.35 

6581  1 

62.38   1 

0325  1 

62.41 

4070 

62.43 

7X17 

62.4(1 

»5''5 

62.48 

5315 

62.51 

9066 

62.54 

2X19  i 

62.56 

6573 

62.59 

0329 

62.61 

4087 : 

62.64 

7846  ! 

62.67 

1607 

61.69 

5369 

62.72 

9133 

62.75 

2899 

62.77 

6666 

62.80 

"435 

62.82 

4205 

62.85 

7977 

62.88 

1750 

62.90 

55*5 

62.93 

9302 

62.96 

3080 

62.98 

6860 

63.01 

0641 

63.04 

4424 

63.06 

8209 

63.09 

«';95 

63.12 

57i<3 

63.14 

957* 

63.17 

3363 

63.19 

715s 

<    63. '.Z 

0949 

.  63.25 

4745 

I  63.27 

8542 

'   63-30 

2341 

'  63.33 

6142 

OOAJ. 

1  63-35 

374« 
7553 

(  I  360 
I  5>68 
1  8978 
I  2790 
I  6603 

0418 
4235 

8°53 
1873 

5695 

9518 

'  "^» 
;  7'68 

0096 

4826 

^  8657 


63-4> 
63.44 

63.46 
63.49 
63.52 
63-54 
63-57 

63.60 
63.62 
63.65 
63. 6X 
63.70 

63-73 
63.76 

63.79 
63. Si 
63.84 

63.87 


i 

V, 

132 

0 

133 

0 
Dlff.  1". 

134 

I.-kM. 

0 
1)1  IT.  1". 

135 

Ion  .M. 

Dlff.  1".  ' 

log  M. 

Dlff.  1". 

'    0 

2.654  8657  1 

63.87 

2.678  1547  ■ 

65-53 

1.702  0562 

67.27 

1.726  5990 

69.09 

1 

.655  2490 

63.89 

.678  5480 

65.5b 

.702  4600 

67.30 

.727  0137 

69.12 

'Z 

.655  6324  1 

63.92 

.678  94<4  1 

65.59 

.702  8638 

'1:^:11 

.727  A285 

•7*7  8435 
.728  25^7 

69.15 

u 

.656  0160 

63.95 

.679  3350 

65.61 

.703  2679 

69.19 

1    * 

.656  3998 

63.97 

.679  7288  1 

65.64 

.703  6721 

67-39 

69.22 

i   5 

2.656  7837 

64,00 

2.680  1227 

65.67 

2.704  0766  j 

67.42 

1.728  6741 

69.25 

1   <^ 

.657  1678 

64.03 

.680  5168  1 

65.70 

.704  4812  , 

67-45 

.729  0897 

69.28 

7 

.657  5521 

64.06 

.680  9111 

65-73 

.704  8860 

67.48 

■7-')   5055 

69.31 

8 

•657  9365 

64.08 

.681  3056 

65.76 

.705  2909 

67.5' 

.719  9215 

6934   i 

I) 

.658  3211 

64.11 

.681  7002 

65-79 

.705  6961 

67-54 

.730  3376 

69-37   1 

'  10 

2.658  7058 

64.14 

2.682  0950 

65.81 

2.706  1014 

67-57 

2.730  7539 

69.40   1 

i  11 

.659  0907 

64.17 

.682  4900 

65.84 

.706  5069 

67.60 

.731  1705 

69.44 

vz 

.659  A758 
.659  861 1 

64.19 

.681  8851 

65.87 

.706  9126 

67.63 

•73'  5^*7^ 

69.47 

13 

64.22 

.683  2804 

65.90 

.707  3184 

67.66 

.732  0041 

69.50 

14 

.660  2465 

64.25 

.683  6759 

65-93 

.707  7244 

67.69 

.732  4212 

69.53 

!  15 

2.660  6320 

64.28 

2.684  0716 

65.96 

2.708  1307 

67.72 

2.732  «3«5 

69.56 

16 

.661  0178 

64.30 

.684  .674 
.684  8634 

65.99 

.708  5371 

67-75 

•733  2559 

69.59 

17 

.661  4037 

64-33 

66.01 

.708  9436 

67.78 

•733  6736 

69.62 

1  18 

.661  7897 

64.36 

.685  2596 

66.04 

.709  3504 

67.81 

•7  34  0914 

69.66 

19 

.662  1760 

64.38 

.685  6559 

66.07 

.709  7573 

67.84 

•734  5°94 

69.69   J 

20 

2.662  5623 

64.41 

2.686  0524 

66.10 

2.710  1645 

67.87 

i-734  9*77 

69.72   ' 

21 

.662  9489 

64.44 

.686  4491 

66.13 

.710  5718 

67.90 

•735  3461 

69-75   , 

22 

.663  3356 

64.47 

.686  8460 

66.16 

.710  9792 

67.93 

•735  7647 

69.78 

23 

.663  7225 

64.49 

.687  2430 

66.19 

.711  3869 

67.96 

.736  1835 

69.81   1 

24 

.664  1096 

64.52 

.687  6402 

66.22 

.711  7947 

67.99 

.736  6025 

69.85   ! 

25 

2.664  4968 

64.55 

2.688  0376 

66.25 

2.712  2028 

68.02 

2.737  0216 

69.88   1 

20 

.664  8842 

64,57 

.688  4352 

66.27 

.712  61 10 

68.05 

•737  44«° 

69.91 

27 

.665  2717 

64.60 

.688  8329 

66.30 

.713  0194 

68.08 

•737  X605 

69,94 

28 

.665  6594 

64-63 

.689  2308 

66.33 

•713  4»79 

68.11 

.738  2803 

69.97 

29 

.666  0473 

64.66 

.689  6289 

66.36 

.713  8367 

68.14 

.738  7002 

70.00 

30 

2.666  4354 

64.69 

2.690  0272 

66.39 

2.714  2456 

68.17 

2.739  1203 
•7  39  5406 

70.04 

31 

.666  8236 

64.72 

.690  4256 

66.42 

•7'4  6547 

68.20 

70.07 

32 

.667  2120 

64-74 

.690  8242 

66.45 

.715  0640 

6S.23 

•739  9612 

70.10 

33 

.667  6005 

64.77 

.691  2230 

66.48 

•7>5  4735 

68.26 

.740  3819 

70.13 
70.16 

34 

.667  9892 

64.80 

.691  6219 

66.51 

.715  8832 

68.29 

.740  8027 

35 

2.668  3781 

64.83 

2.692  0210 

66.54 

2.716  2930 

68.32 

2. ■'41  2238 

70.20 

3G 

.668  7672 

64.86 

.692  4203 

66.56 

.716  7031 

68.35 

.741  6451 

70,23 

37 

.669  1564 

64.88 

.692  8198 

66.59 

.717  1133 

68.38 

.742  0666 

70,26 

38 

•669  5457 

64.91 

.693  2194 

66.62 

.717  5237 

68.41 

.742  4882 

70.29 

39 

,669  9353 

64.94 

-693  6193 

66.65 

.717  9342 

68.44 

.742  9101 

70.32 

40 

2.670  3250 

64.97 

2.694  0193 

66.68 

2.718  3450 

68.48 

2.743  33*' 

70.36 

41 

.670  7149 

65.00 

.694  4194 
.694  8198 

66.71 

.718  7560 

68.51 

•743  7543 
•744  1768 

70.39 

42 

.671  1050 

65.02 

66.74 

.719  1671 

68.54 

70.42 

43 

.671  491:2 

65.05 

.695  2203 

66.77 

.719  5784 

68.57 

•744  5994 

70-45   S 

44 

.671  8856 

65.08 

.695  6210 

66.80 

.719  9899 

68.60 

.745  0222 

70.48 

45 

2.672  2761 

65.11 

2.696  0219 

66.83 

2.720  4016 

68.63 
68.66 

2.745  445* 

70.52 

46 

.672  6668 

65.13 
65.16 

.696  4229 

66.86 

.720  8135 

-745  8684 

70-55 

47 

.673  0577 

.696  8242 

66.89 

.721  2255 

68.69 

.746  2918 

70.58 

48 

.673  4488 

65.19 

.697  2256 

66.92 

.721  6377 

68.72 

.746  7154 

70.61 

49 

.673  8400 

65.22 

.697  6272 

66.95 

.722  0502 

68.75 

•747  1391 

70.65   1 

50 

2.674  2314 

65.25 

2.698  0289 

66.97 

2.722  4628 

68.78 

2.747  5631 

70.68 

51 

.674  6230 

65.2§ 

.69S  4308 

67.00 

.722  87<;6 

68.81 

•747  9873 

70.71   1 

52 

•675  o«47 

65.30 

.698  8330 

67.03 

.723  2885 

68.84 

.74?  4n6 

70.74   1 

53 

.675  4066 

65.36 

•699  2353 

67.06 

.723  7017 

68.88 

.748  8362 

70.78 

54 

.675  7987 

•699  6377 

67.09 

.724  1150 

68.91 

.749  2609 

70.81 

53 

2.676  1909 

6539 

2.700  0404 

67.12 

2.724  5286 

68.94 

2.749  6859 

70.84 

56 

.676  5833 

65.42 

.700  4432 
.700  8462 

67.15 

.724  9423 

68.97 

.750  mo 

70.87 

1  57 

■676  9759 

65.44 

67.18 

.725  3562 

69.00 

.750  5364 
.750  9619 

70.90 

58 

•677  3687 

65.47 

.701  2494 

67.21 

.725  7703 

69-03 
69.06 

70.94 

59 

.677  7616 

65.50 

.701  6527 

67.24 

.726  1846 

.751  3876 

70.97 

60 

2.678  1547 

65-53 

2.702  0562 

67.27 

2.726  5990 

69.09 

2.751  8135 

71.00 

599 


TABLE  VI. 

For  tiiitlinK  llic  Triio  Aiiomiily  or  lliv  Tiiiif  tVnin  tin-  I'i'riliclioii  in  a  I'liniliolic  Orhit. 


V. 
0' 

lot 

136" 

« 

137 

138° 

13£ 

l-K  .M. 

i.831  8224 

Din.  1". 
77^31 

M. 

Dili.  I". 
71.00 

Inn 

2.777 

M. 

73" 

inn.  1". 
73.01 

loK  M. 

2.804  3895 
.804  8403 

Dirr.  1". 
75.11 

X.75I 

Hi  J 5 

1 

•7<;i 

2396 

71.03 

•778 

1703 

73.04 

75.'.^ 
75.18 

.832  »86i 
.832  7506 

77^35 

•z 

•751 

6659 

71.07 

.778 

6087 

73^o7 

.805  2912 

77^39 

a 

•753 

0925 

71.10 

■779 

0472 

7311 

•8oi  7424 

75.21 

.833  2151 

77^43 

4 

•753 

5192 

71.13 

•779 

4859 

73'4 

.806  1938 

75-15 

.833  6798 

77^47 

5 

^•753 

9461 

71.17 

2.779 

9249 

73.'8 

2.806  6454 

7  5^i9 

2.834  "447 

77.50 

<) 

•754 

373* 

71.20 

.780 

1641 

8034 

73^21 

.807  0973 

75^3i 

.834  6098 

77.54 

77.58 

7 

•754 

H004 

71.23 
71.26 

.780 

73.*4 
73-18 

.807  5493 

75.36 

.835  0752 

H 

•755 

2279 

.781 

2430 

.808  0016 

75.40 

•835  5408 
.836  0066 

77.62 

U 

•755 

6556 

71.30 

.781 

6828 

73.31 

.808  4541 

75^43 

77.66 

10 

^•75'> 

0X3; 

7i^-13 
71.36 

2.782 

1228 

73^35 

2.808  9068 

75^47 

2.836  4727 

77.69 

11 

.756 

;i  16 

.782 

5630 

7338 

.809  3597 
.809  8128 

75.50 

.836  9390 

11-11 

lU 

•756 

9399 

71.40 

.783 

0034 

73-42 

75.54 
75^58 

•837  4055 

11-11 

i:t 

•757 

3683 

7i^4; 
71.46 

•783 

4440 

73-45 

.810  2662 

.837  8722 

TI.U 

11 

•757 

7970 

•783 

8848 

73-49 

.810  7197 

75^6 1 

•838  3391 

77.85 

15 

1.758 

2259 

71.49 

2.784 

3158 

73^51 

73.56 

2.81,  ,735 

75^65 

2.838  8064 

77.89 

10 

.75S 

6549 

7i^51 
71.56 

.784  7('7i 

.811  6275 

75.69 

•839  1738 

77^9i 

17 

•759 

0S42 

•785 

2085 

73-59 

.812  0817 

75.72 

•839  74'4 

77^96  ; 

IH 

•759 

5»37 

7»^59 

.785 

6502 

73.63 
73^66 

.812  5362 

75.76 

.840  2093 

78.00 

lU 

•759 

9433 

71.63 

.786 

0920 

.812  9908 

li-T) 

.840  6774 

78.04 

uo 

2.760 

3732 
8032 

71.66 

2.786 

534' 

9764 

73^7o 

i.813  4457 

75^83 

2.841  '458 

78.08 

'Zl 

.760 

71.69 

.786 

73^7; 
73.76 

.813  9008 

75-87 

.841  6144 

78.11 

'i'Z 

.7fii 

1335 

7^73 
71.76 

.787 

4189 

.814  3561 

75.90 

.842  0832 

78.15 

'Z-A 

.761 

6639 

.787 

8615 

73.80 

.814  8117 

75.94 
75.98 

.842  5522 

78.19 

'Z\ 

./bz 

0946 

7  ••79 

.788 

3044 

73-83 

.815  2674 

.843  0215 

78.23 

25 

1.7(12 

5*55 

71.83 

2.788 

7476 

73^87 

2.815  7134 
.816  1796 

76.01 

1.843  4909 

78.27 

20 

.762 

9565 

71.86 

•789 

1909 

73.90 

76.05 

.843  9607 

78.31 

27 

.763 

3878 
8192 

71.89 

.789 

6344 

73-94 

.816  6360 

76.09 

.844  4306 

78.35 

28 

.763 

71-93 
71.96 

.790 

0781 

7  3-97 

.817  0927 

76.12 

.844  9008 

78.38 

20 

.764 

2509 

.790 

5221 

74.01 

.817  5495 

76.16 

.845  3712 

78.42 

30 

2.764 

6827 

71.99 

2.790 

9662 

74.04 

2.818  0066 

76.20 

2.845  8419 

78.46 

31 

.765 

1148 

72.03 

.791 

4106 

74.08 

.818  4639 

76.23 

.846  3128 

78.50 

:i2 

.765 

5470 

72.06 

.791 

8552 

74." 

.818  9214 

76.27 

.846  7839 

78.54 

33 

.765 

9795 

72.09 

.792 

3000 

74-' 5 

.819  3792 

76.31 

•847  2553 

78.58 

34 

.766 

4121 

72.13 

.792 

7450 

74.18 

.819  8371 

76.34 

.847  7268 

78.62 

35 

2.766  8450 

72.16 

2.793 

1902 

74.22 

2.820  2953 

76.38 

2.848  1986 

78.66 

36 

.767 

2781 

72.19 

•793 

6356 

74.25 

.820  7537 

76.42 

.848  6707 

78.69 

37 

.767 

7113 

72.23 

•794 

0813 

74.29 

.821  2123 

76.46 

•849  '43° 

78.73 

38 

.768 

I44» 
5784 

72.26 

•794 

5271 

74-32 

.821  6712 

76.49 

.819  6155 

78.77 

30 

.768 

72.29 

•794 

9731 

74.36 

.87,2  1302 

76.53 

.850  0882 

78.81 

40 

2.769 

0123 

72.33 

^•795 

4194 

74.40 

2.8:dJ,  ,895 

76^57 

2.850  5612 

78.85 

41 

.769 

4464 

72.36 

•795 

8659 

74-43 

.ti\   0(91 

76.60 

•85 '  0344 

78..S9 

42 

.769 

8806 

72.39 

.796 

•jl26 

74^47 

.9i>,    ;j88 

76.64 

.851  5079 

78.93 

43 

.770 

3151 

72^43 
72.46 

.796 

7595 

74-50 

.■:?  9688 

76.68 

.851  9816 

78.97 

44 

.770 

7498 

•797 

2066 

74-54 

.i,z\   .^289 

76.72 

.852  4555 

79.01 

45 

2.771 

1846 

72.50 

2.797 

6539 

74.58 

2.824  8894 

76.7s 

2.852  9297 

79.05 

40 

•77' 

6197 

72.53 

.798 

1015 

74^6 1 

.825  3500 

76.79 

•853  404' 

79.08 

47 

.772 

0550 

72.56 

.798 

5491 

74.64 

.825  8108 

76.83 

.853  8787 

79.12 

48 

.772 

4905 

72.60 

•798 

9972 

74.68 

.826  2719 

76.87 

•854  3535 

79.16 

40 

.772 

9262 

72.63 

•799  4454 

74.7' 

.826  7332 

76.90 

.854  8286 

79.20 

50 

2.773 

3621 

72.67 

2.799 

8938 

74^75 

2.827  1947 

76.94 

2.855  3040 

79-14   : 

51 

•773 

7982 

72.70 

.800 

3424 

74^79 

•827  6565 

76.98 

•855  7795 

79.28 

52 

•774 

2344 

72.73 

.800 

7912 

74.82 

.828  1185 

77.01 

.856  2553 

79-3J 

53 

•774 

6709 

72.77 

.801 

2402 

74.86 

.828  5807 

77.05 

.856  7314 

79-36 

54 

•775 

1077 

72.80 

.801 

6895 

74.89 

.829  0431 

77.09 

.857  2077 

79.40 

55 

^•775 

5446 
9817 

72.84 

2.802 

1390 

74-93 
74.96 

2.829  5058 

77.13 

2.857  6842 

79-4-t 

50 

•775 

72.87 

.802 

5886 

.829  9686 

77.16 

.858  160.7 

79.48  ; 

57 

.776 

4190 
8565 

72.90 

.803 

0385 
4886 

75.00 

.830  4317 

77.20 

.858  6379 

79-5; 

58 

.776 

72.94 

.803 

75^04 

.830  8951 

77-14 
77-18 

.859  1151 

79.56 

50 

•777 

2942 

72.97 

.803 

9390 

75.08 

.831  3586 

.859  5926 

79.O0 

GO 

2.777 

7322 

73.01 

2.804 

3895 

75.11 

2.831  8224 

77.32 

2.860  0703 

79.64 

\ 

600 

■  • 

liolic  Orhit. 


5612 
0344 

5079 
9816 

4555 


3040 
7795 
^555 
73>4 
2077 

6842 

leoT 
6379 

5926 
0703 


lllff.  I". 

77.3» 

77-15 

7;-i9 

77-45 
7  7-47 

77-5° 
77-54 
77-55 
77.62 
77.66 

77.69 

77-73 
77-77 
77.X1 

77-«5 

77.89 

77-9» 
77.96 

78.00 
78.04 

78.08 
78.11 
78.15 
78.19 
78.23 

78.27 
7«-1" 
7«-35 
78.38 
78.42 

78.46 
78.50 

7«-54 
78.58 

78.62 

78.66 
78.69 

7«-73 
78-77 
78.81 

78.85 
78.89 
78.93 
78.97 
79.01 

79.08 
79.12 
79.16 
79.20 

79-'4 
79.28 

79-3J 
79.36 
79.40 

79-44 
79.48 

79-55 
79.56 
79. Co 

79.64 


TABLE  VI. 

l''(ir  limliin,'  the  Tnio  .Vriornnlv  or  lln-  Tiim-  li-cuii  tlio  IVrilu-linn  in  n,  Parabolic  Orhit. 


1  ♦'• 

14C 

)" 

141 

loK  M. 

0 
DIPT.  1". 

14S 

l"Kll. 

5^ 

Dinr.  1". 

!•>« 

142 

M. 

r : 

DIff.  1". 

Diir.  I". 

0 

2.860  0703 

79.64 

79.6)5 

..8«9  1754 

82.08 

2.919  iBjI 

84.65 

1.950 

1420 

87-37 

1 

.860  5482 

.889  6680 

82.12 

.919  6911 

84.70 

■  950 

6664 

87.4,   [ 

'Z 

.861  0264 

79.7» 

.890  1609 

82.16 

.920  1994 

84.74 

■95" 

1910 

87.46 

i  :> 

.861  5048 

79.76 

.890  6540 

81.20 

.920  7080 

84.78 

•951 

7' 59 

87,50 

'1 

.X61  9835 

79.80 

.891  1473 

82.25 

.921  2169 

84.83 

.952 

1411 

87-55 

n 

2.8(-i  4624 

79.8)5 

2.X91  6409 

82.29 

2.921  7260 

84,87 

2.952 

7665 

87.60 

» 

.861  9415 

.892  1348 

81.33 

-9"  »353 

84.91 

•95  3 

2923 

87,65 

7 

.863  4209 

79.92 

.892  6289 

82.37 

.922  74;;o 

84.96 

-953 

8183 

87.69 

N 

.863  9005 

79.96 

.893  1233 

82.41 

.923  2549 

85.01 

-954 

344" 

87-74 

11 

.864  3803 

80.00 

.893  6179 

82,46 

.923  7650 

85.05 

-954 

8711 

87.79 

10 

1,864  8604 

80.04 
80.08 

2.894  1127 

82.50 

i.924  2755 

85.10 

1-955 

3980 

87.81 
87.88 

11 

.86^  3408 

.894  6078 

82.54 
82.58 

.924  7861 

85.14 

-955 

92^1 

Vi 

.861;  8213 

80. 1  2 

.895  1032 

.925  2972 

85.18 

.956 

4515 

87-93   ! 

1:1 

.866  3021 

80.16 

.895  5989 

82.63 

.925  8084 

85.23 

,956 

9802 

ti-''"^ 

II 

.866  7832 

80.20 

.896  0948 

82.67 

,926  3199 

85.27 

-957 

5082 

88,o2 

15 

2.867  2645 

80.24 

2.896  5909 

82.71 

Z.926  8317 

85.32 

2.958 

0365 

88.07 

10 

.867  7460 

80.28 

.897  0873 

82.75 

•917  3437 

85.36 

.958 

5651 

88.11 

17 

.868  2278 

80.32 

.897  5839 

82.79 

.927  8560 

85.41 

-959 

0939 

X8.16 

IH 

.868  7098 

80.36 

.898  0808 

82.84 
82.88 

.928  36S6 

85-45 

•959 

6230 

88.21 

Ml 

.869  1921 

80.40 

.898  5780 

.928  8814 

85.50 

.960 

1524 

88.26 

'ZO 

2.X69  6746 

80.44 
80.48 

2.899  071  : 

82.92 

1.919  3945 

85-54 

2.960 

6821 

^!!-3° 

'Zl 

.870  1573 

.899  5730 

82. 96 

.919  9079 

85.59 

.961 

2120 

^^35 

•Z'Z 

.870  6403 

80.52 

.900  0709 

83.CI 

.930  4116 

85.63 
85.68 

.961 

7423 

88.40 

•ZA 

.871  1235 

80.56 

.900  5691 

83.05 

-93°  9355 

.962 

2728 

88.45    : 

•Zi 

.871  6070 

80.60 

901  0675 

83.09 

■93'  4497 

85.71 

.962 

7036 

88.49 

'zrt 

2.872  0907 

80.64 

1.901  5662 

83.13 

1.931  9641 

85-77 

2,963 

3347 

!!-5+ 

•za 

.872  5747 

80.68 

.902  0651 

83.18 

.932  47S8 

85.81 

,963 

8661 

88,, 9 

'Zl 

.873  0589 

80.72 

.902  5643 

83.21 

.932  9938 

85.86 

.964 

3978 

88,64 

'ZH 

•i*73  5433 

80.76 

.903  0638 

83.26 

•93  3  509' 

85-91 

.964 

9»97 

88,68   ' 

a« 

.874  0280 

80.80 

.903  5635 

83.3, 

•9  34  0*47 

85.95 

.965  4620 

88.73 

;io 

2.874  5129 

80.84 

2.904  0635 

83-35 

2-934  S4°5 

85-99 

2.965 

9945 

88.78 

:ii 

.874  9981 

80.88 

.904  5637 

83.39 

-935  0565 

86,04 
86,08 

.966 

5173 

88.83 

A'Z 

.875  4835 

80.92 

.905  0642 

«3-43 

-935  5729 

.967 

0604 

88.87 

AA 

.875  9692 

80.96 

.905  5649 

83.48 

.936  0895 

86., 3 

.967 

5938 

ll-'f"- 

31 

.876  4551 

81.01 

.906  0659 

83.51 

.936  6064 

86.17 

.968 

1275 

88.97 

:i5 

2.876  9413 

81.05 

2.906  5672 

83.56 

2.937  1236 

86.22 

2,968 

66:5 

89.02 

»6 

.877  4277 

8 1 .09 

.907  0687 

83.61 

.937  6410 

86,26 

.969 

1957 

89.07 

37 

.877  9143 

8. .13 

.907  5704 

83.65 

.938  1587 

86.31 

.969 

7303 

89.12 

38 

.878  4012 

8,., 7 

.908  0725 

83.69 

.938  6767 

86.35 

.970 

265. 

89.17 

39 

.878  8883 

81.21 

.908  5748 

«3-74 

•939  >95o 

86.40 

•97° 

8002 

89.11 

10 

2.879  3757 

81.25 

2.909  0773 

83.78 

2.939  7135 

86,45 

2.971 

3356 

89.26 

41 

.879  8633 

81.29 

.909  5801 

83.82 

.940  2323 

86.49 

•97' 

8713 

89,31 

VZ 

.880  3512 

81.33 

.910  0832 

83.87 

-94°  75 > 4 

86,54 

.972 

4073 

89.36 

43 

.880  8393 

81.37 

.910  5865 

83.91 

.941  2708 

86.58 

.971 

9436 

89.-40 

44 

.881  3277 

81.42 

.911  0901 

83-95 

.941  7904 

86.63 

■973 

4801 

89.4s 

45 

2.881  8163 

81.46 

2.911  5940 

83.99 

2.942  3103 

86,67 

1.974 

0170 

89,50 

4» 

.882  3052 

Si. 50 

.912  0981 

84.04 

.942  8305 

86.71 

•974 

554' 

89-55    : 

47 

.882  7943 

81.54 

.912  6024 

84.08 

•943  35 «° 

86.77 

•975 

0916 

89,60 

48 

.883  2837 

81.58 

.913  1070 

84.13 

•943  8717 

86.81 

■975 

6193 

89.05 

4« 

-«83  773  3 

81.62 

.913  6119 

84.17 

-944  3927 

86.86 

.976 

1673 

89.69 

50 

2.884  ='*'3i 

81.66 

2.914  1171 

84.22 

2.944  9140 

86.90 

1,976 

7056 

89.74 

51 

.884  7532 

81.70 

.914  6225 

84.26 

•945  43  5  5 

86.95 

•977 

1442 

89,79 

5« 

.885  2436 

81.75 

.915  1282 

84.30 

•945  9574 

87.00 

•977 

7831 

89.84 

53 

.885  7342 

81.79 

.915  6341 

84.34 

-946  4795 

87.04 

.978 

3223 

89.89 

;  54 

.886  2251 

81.83 

.916  1403 

84.39 

.947  0019 

87.09 

■978 

8618 

89,94 

'  55 

2.886  7162 

81.87 

2.916  6468 

84.43 

*-947  5^45 

87.11 

2.979  4015 

89.99 

5U 

.887  2075 

81.91 

•9«7  1535 

84.48 

.948  0475 

87,18 

•979 

9416 

90.03 

57 

.887  6991 

81.95 

.917  6605 

84.52 

.948  5707 

87.23 

.980 

4820 

90.08 

58 

.888  1910 

81.99 

.918  1678 

84.56 

-949  °94a 

87.27 

.981 

1226 

90.15 
90.18 

59 

.888  6831 

82.04 

.918  675J 

84.61 

.949  6180 

87.3* 

.981 

6636 

(K) 

2.889  1754 

82.08 

2.919  1831 

84.6s 

2.950  1420 

87^37 

1.982 

1048 

90.a3 

1 

601 

TABLE  VI. 

Tor  finding  flic  Tnte  Anomaly  or  the  Time  Iroiu  the  Perihelion  in  a  Paraboli''  Orbit. 


t 
i   ^- 

i 

144° 

145° 

146° 

147° 

log 

M. 

1048 

Diff.  1". 

lofi 

M 

Uiff.  1". 
93.26 

log 

M. 

I)ilT.  1". 

loK  M, 

DilT,  I". 

0 

2.982 

90.23 

3.015 

1281 

3,049 

2733 

96.47 

3.084 

6070 

99,87    ' 

1 

.9X2 

'^('■i 

90.28 

.015 

6X7X 

93^3> 

,049 

8522 

96.52 

.0X5 

2064 

99,92 

2 

.9X3 

1XX2 

90.33 

.016 

247X 

93-36 

.050 

43'5 

96.58 

.085 

8061 

99.9S    ■ 

3, 

.9X3 

7303 

go.  3  8 

.016 

X082 

93-4^ 

.051 

01 1  2 

96.63 

.0X6 

4062 

ir^  ni 

4' 

.9X4 

2727 

90.43 

.017 

36X8 

93-47 

.051 

5911 

96.69 

.087 

0066 

100.10 

1 

5 

2.9X4 

8.54 

90.48 

3.017 

9298 

93.52 

3.052 

1714 

96,74 

3.087 

60-,  3 

100,16 

G 

.9X.; 

3584 

90-53 
90.158 

.oiS 

4911 

93^57 

.052 

7520 

96,80 

.oXX 

20X5 

100,22 

7 

.9X,- 

9017 

.019 

0526 

97.(12 

-^53 

3329 

96.85 

.088 

8099 

100.2S 

8 

.9X0 

4453 

90.63 

.019 

6145 

93.68 

.053 

9142 

96.91 

.089 

411X 

100.33 

0 

.9X6  9X9.. 

90.67 

.020 

1768 

93^7  3 

.054  4959 

96.96 

.090 

0140 

100,39  J 

10 

2.987 

5334 

90.72 

3.020 

7393 

93^78 

3^055 

0778 

97,01 

3.090 

6165 

100,45 

11 

.9XX 

0779 

90.77 

.021 

3021 

93.X3 

.055 

6601 

97.07 

.091 

2194 

100.51 

Vi 

.9XX 

6227 

9Q.X2 

.021 

8653 

93.89 

.056 

2427 

9/^>3 

.091 

8226 

100.57 

i:i 

.989 

1678 

90.X7 

.022 

42XX 

93-94 

.056 

8256 

97.19 

.092 

4262 

100.63 

14 

.989 

7132 

90.92 

.022 

9926 

93^99 

.057 

4089 

97.24 

.093 

0302 

100.69 

15 

2.990 

2589 

90.97 

3.023 

5567 

94.04 

3.057 

9925 

97.30 

3-093 

6345 

100.75 

10 

.990 

8c49 

9 1 .02 

.024 

1211 

94.10 

.05X 

5765 

97^35 

•094 

2392 

100, Xl 

17 

•99  > 

3512 

9-  07 

.024 

r-!?59 

94^«5 

,059 

1608 

97^4i 

.094 

8.  -2 

100.87 

18 

.991 

8977 

91.12 

.025 

2509 

94.20 

,059 

7454 

97^47 

.095 

4496 

100.93 

10 

.992 

4446 

91,17 

.025 

8163 

94.26 

,060 

3304 

97.52 

.096 

0553 

100,98 

20 

2.992 

9918 

91.22 

3.026 

3820 

94.31 

3,060 

9157 

97.58 

3.096 

66,+ 

101.04 

21 

•993 

5  393 

91.27 

.026 

94X0 

94.36 

,061 

5o«3 

97-63 

•097 

2678 

101,10 

22 

•994 

0X71 

91.32 

.027 

5  "43 

94.41 

.062 

0873 

97.69 

.097 

8746 

101.16 

23 

•99+ 

6351 

9«^37 

.028 

oXio 

9447 

.062 

6736 

97^75 

.098  4818 

IOI.12 

24 

•995 

1835 

91.42 

.02X 

6479 

94.52 

.063 

2602 

97.80 

-"99 

0893 

101.28 

25 

^•99  5 

7322 

91.47 

3.029 

2152 

94^5" 

3.063  8472 

97.86 

3.099 

6972 

101.34 

26 

■99'' 

2X12 

91.52 

.029 

7S2S 

94.6 1 
9.;  6,? 

.064 

4345 

97,91 

.100 

3054 

IOI..).0 

27 

•99" 

8305 

9^57 

.030 

3507 

.065 

0222 

97-97 

.100 

9140 

101.46 

28 

•997 

3X01 

91.62 

.030 

9190 

/4^:3 

.06  c 

6101 

9X.03 

.101 

5230 

101.52   ' 

29 

•997 

9300 

91.67 

.031 

4875 

"■+•79 

.066 

1985 

98.0X 

.102 

1323 

101. 58 

30 

7.99X  4X02 

91.72 

3.032 

0564 

94.84 

3.066 

7872 

98,14 

3,102 

7420 

101.64 

31 

•999 

0307 

91.77 

.032 

6256 

94.89 

.067 

3762 

98,20 

.103 

3520 

101.70 

32 

•999 

5X15 

91. X2 

.033 

1951 

9494 

.067 

9''5  5 

98,21; 

.103 

9624 

101.76 

33 

3.000 

rf-'' 

91.87 

.033 

7650 

95.00 

.06S 

5552 

98-31 

.104 

5732 

101.82 

34 

.000 

6X40 

91.93 

.034 

3351 

95.05 

.069 

'45  3 

98.37 

.105 

1843 

101.88    ■ 

1 

35 

3.001 

2357 

91.98 

3^034 

9056 

95.11 

3.069 

7357 

9X.42 

3.105 

7958 

101.94 

30 

.001 

7877 

92.03 

.035 

4704 

95.-6 

.070 

3264 

9X.4X 

.106 

4076 

102.00 

37 

.002 

3400 

92. oX 

.036 

0475 

95.22 

.070 

917-1 

98,54 

.107 

0198 

102.07 

38 

.00  7. 

X926 

92.13 

.036 

6190 

95.27 

.07. 

50X8 

98.60 

.107 

6324 

102.13 

3« 

.003 

4456 

92.18 

•037 

1908 

95,32 

.072 

1006 

98.65 

.108 

24:4 

102.19   , 

40 

3.003 

9988 

92.23 

3^°37 

7629 

95.38 

3.072 

6927 

98,71 

3.108 

8587 

102.25 

41 

.004 

55^3 

92.28 

.038 

3353 

95^43 

•073 

2851 

98.77 

.109 

4723 

102.31 

42 

.005 

1062 

91^33 

.038 

90X0 

95.4X 

.073 

8779 

98, X2 

.1 10 

0864 

102.37 

43 

.005 

6603 

92.38 

•039 

4811 

95^54 

.074 

4710 

9X,88 

.110 

7008 

102.43 

44 

.006 

214X 

92.44 

.040 

0545 

95.60 

.075 

0645 

98.94 

,111 

3«55 

102.49 

45 

3.006 

7696 

92.49 

3.040 

6282 

95^65 

3-°75 

65S3 

99,00 

3,111 

9306 

102.55 

40 

.007 

3246 

92.54 

.041 

2023 

95.70 

.076 

2524 

99,05 

.112 

5461 

102.61 

47 

.007 

XSoo 

9^59 

.041 

7767 

95.76 

.076 

8469 

99.11 

•'»3 

1620 

102.67 

48 

.ooX 

4357 

92.64 

.042 

35«4 

95.8. 

.077 

441 X 

99,17 

.113 

7782 

102.73 

49 

.008 

9917 

92.69 

.042 

9264 

95,86 

.078 

0370 

99,23 

.114 

3948 

102. Xo 

50 

3.009 

5480 

92.74 

3.043 

5017 

95.92 

3.078 

6325 

99.28 

3. 115 

0118 

102.86 

51 

.010 

1046 

92.79 

.04.., 

0774 

95^97 

.079 

2284 

99-34 

.115 

6291 

102,92 

52 

.010 

foi5 

92.85 

.044 

6534 

96.03 

.079 

8246 

99,40 

.116 

2468 

102,98 

53 

.011 

2188 

92.90 

.045 

2297 

96.08 

.080 

4212 

99.46 

.116 

8649 

103,04 

54 

.oil 

7763 

92.95 

.045 

8064 

96,14 

.081 

ri8l 

99.52 

.117 

4833 

103.10 

55 

3.012 

3342 

93.00 

3.146 

3834 

96.19 

3.081 

6154 

99-57 

3. 118 

1022 

103, '6 

50 

.012 

8923 

93.05 

.o;6 

9607 

96,25 

.0X2 

2130 

99.63 

.uS 

7213 

103,25 

57 

.013 

4508 

93.10 

.047 

5383 

96.30 

.oX-.', 

8110 

99.69 

.119 

3409 

103,29 

58 

.014 

0    ;6 

93.16 

.04X 

1163 

96.36 

.083 

4093 

99-75 

.119 

9608 

103-3; 

59 

.014 

5687 

93.21 

.048 

6946 

96,41 

.084 

00X0 

99-81 

.120 

5811 

103.41 

GO 

3-o«5 

1281 

93.26 

3.049 

2733 

96.47 

3.084  6070 

99-87 

3.121 

2018 

,03.48 

602 


)olii'  Orbit. 


TABLE  VI. 

For  finding  tlie  True  Anoinnly  or  the  Time  fnini  the  Poriholion  in  a  Parabolic  (^rbit. 


DilT.  1". 

99.87 
99.9- 
99. 9S 

IC'-'  n\ 
100.10 


6o',  3  1 

100.16 

2085  ',    100.22 

8099   100.2S 

4118  :  100.33 

0140  i  100.39 

6165  ;  100.45 

2194  1  100.51 

8226  ,  100.57 

4262   100.63 

0302  1  100.69 

6US 

100.75 

2392  I 

100. Si 

8j.;2  i 

100.87 

4496 

100.93 

°553 

100.98 

6614 

101.04 

2678  1  101. 10 

8746  i  101.16 

4818   101.22 

0893  1  loi.2!S 

6972  1  101.34 

3054  1  101.40 

9140   101.46 

5230  1  101.52 

1323  '  101. 58 

7420   101.64 

3520  1  101.70 

9624 

101.76 

573^ 

101.82 

1843 

101.88 

7958  !  101.94 

4076  j  102.00 

0198  '  102.07 

6324 

102.13 

24:4 

102.19 

8587 

102.25 

4723 

102.31 

0864  i  102.37 

)  7008  i  102.43 

3«55 

102.49 

9306 

102.55 

-  546' 

102.61 

I  1620 

102.67 

,  7782 

102.73 

^  394» 

101.80 

5  0118  ■  102.86 

5  6291  .  102.92 

5  2468  1  102.9S 

ft  8649  1  103.04 

7  4«33 

103.10 

8  1022 

103. '6 

S  7213 

103.13 

9  3409 

103.29 

9  9608 

103.35 

0  5811 

103.41 

I  2018  {  103.48 

V. 
0' 

148° 

149° 

150° 

151 

lot?  M. 
3.239  3820 

0       ' 
1 

1«K 

M. 

I)i(T.  1". 
103.48 

log  M. 

Diff.  1". 

loK 

M. 

I)ilT.  1". 
111.41 

Diff.  1".  j 
115-77  < 

3.I2I 

2018 

3.159  1367  I 

107.31 

3.198  4984  I 

1 

.121 

8228 

103.54 

.159  y^o^ 

107.38 

•'99 

1671 

111.48 

.240  0768 

115.85 

'i 

.122 

4AA2 
0660 

103.60 

.160  4253 

107.45 

■'99 

8361 

III. 55 

.240  7722 

115.92 

3 

•'i3 

103.66 

.161  0702 

107.51 

.200 

5056 

111.62 

.241  4680 

116.00  ' 

4 

•«J3 

6882 

103.72 

.161  7154 

107.58 

.201 

'755 

111.69 

.242  1642 

116.08  1 

1   5 

3«24 

3107 

103.79 

3.162  3611 

107.65 

3.2)1 

8459 

111.76 

3.242  8608 

116.15  ■ 

1   « 

.124 

9336 

103.85 

.163  0072 

107.71 

.2  12 

5166 

111.83 

.243  5580 

116.23  ! 

1   7 

.125 

5569 

103.91 

.163  6J36 

107.78 

.2,3 

1878 

111.90 

•244  255''^ 

116.30  ] 

8 

.126 

1805 

103.97 

.164  3005 

107.85 

.203 

8594 

111.97 

•244  9536 

116.38  : 

1   1> 

.126 

8045 

104.04 

.164  9478 

107.91 

.204 

53'5 

1 12.04 

.245  6521 

116.45 

lU 

3.127 

4289 

1 04. 1 0 

3'"5  5955 

107.98 

3.205 

2040 

112.11 

3.246  3511 

116.53  1 

11 

.128 

0537 

104.16 

.166  2435 

108.04 

.205 

8769 

112.18 

.247  0505 

116.61 

;  1'^ 

.128 

6789 

104.22 

.166  8920 

108.11 

.206 

5502 

112.26 

■247  7503 

116.68 

;  13 

.129 

3044 

IOJ.29 

.167  5409 

108.18 

.207 

2^39 

112.33 

•248  4507 

116.76 

1  14 

.129 

9303 

104.35 

.168  1901 

108.25 

.207 

8981 

1 12.40 

•2+9  •5'S 

116.84 

15 

3-'3° 

5566 

104.41 

3.168  8398 

108.31 

3.208 

5727 

112.47 

3.249  8i27 

116.91 

16 

.131 

1833 

104.48 

.169  4899 

108.38 

.209 

2478 

112.54 

.250  5544 

1  16.99 

17 

.131 

8103 

104.54 

.170  1404 

108.45 

.209 

9232 

1 12.61 

.251  2566 

117.07 

18 

.132 

4377 

104.60 

.170  7913 

108.51 

.2IO 

599' 

112.69 

.251  9592 

117.14 

lU 

.133 

065s 

104.67 

.171  4426 

108.58 

.211 

2755 

112.76 

.252  6623 

117.22 

20 

3'33 

6937 

104.73 

3.172  0942 

108.65 

3.211 

9522 

112.83 

3^253  3658 

117.30 

21 

•«34 

3223 

104.79 

.172  7463 

108.72 

.il2 

6294 

1 12.90 

.254  0698 

"7-37  i 

1  22 

•34 

9512 

104.86 

•'73  39XX 

108.78 

.213 

3070 

112.97 

•254  7743 

"7^45  i 

i  23 

•135 

5805 

104.92 

•'74  °5'7 

108.85 

.213 

985. 

113.05 

•255  4792 

"7^53 

i  24 

.^36 

2102 

104.98 

•'74  705' 

108.92 

.214 

6636 

113.12 

.256  1846 

117.60 

i  25 

3.136  8403 

105.05 

3,175  3588 

108.99 

3.215 

3425 

113.19 

3.256  8905 

117.68 

!  2G 

••37 

4708 

105.11 

.176  0129 

109.06 

.216 

0219 

113.26 

.257  596*; 

117.76 

i  27 

.138 

1016 

105.17 

.176  (1674 

1 09. 1 2 

.216 

7017 

"3-34 

.258  3036 

117.84 

;  28 

.138 

7329 

105.24 

•'77  3224 

109.19 

.217 

38,9 

113.41 

.2s9  oic; 

117.91 

29 

.139 

3045 

105.30 

•'77  9777 

109.26 

.218 

0626 

113.48 

.259  7186 

117.99 

30 

3-' 39 

9965 

105.36 

3.178  6335 

10^.33 

3.2.  ^^ 

7437 

"3-55 

3.200  4268 

118.07 

31 

.140 

6289 

i°5-43 

•  '79  ^^97 

109.40 

.219 

4252 

113.63 

.2C1  1354 

118.15 

32 

.141 

2616 

105.49 

■'79  94'^i 

109.46 

.220 

1072 

113.70 

.261  8446 

118.23 

33 

.141 

8948 

105.55 

.iSo  6032 

109.53 

.220 

7896 

"3-77 

.262  5542 

118.30 

34 

.142 

5283 

105.62 

i8i  2606 

109.60 

.221 

4724 

113.84 

.263  2642 

118.38 

35 

3143 

1622 

105.68 

3. 181  9184 

109.67 

3.222 

'557 

113.92 

3.263  9747 

11 8. 46 

3C 

•«43 

7965 

105.75 

.182  5766 

109.74 

.222 

8395 

1 1  3.99 

.26^  6857 

118.54 

37 

.144 

43'2 

105.81 

•'X3  2353 

109.81 

.223 

5136 

ll.i  c6 

.265  3972 

118.62 

I    38 

•'45 

0663 
7018 

105.87 

.183  8943 

109.87 

.224 

2081 

114.14 

.266  1091 

118.70 

30 

.145 

105.94 

.184  5538 

109.94 

.224 

8933 

1 14.21 

.266  8216 

118.77 

40 

3.146 

3376 

106.00 

•>  185  2136 

110.01 

3.225 

5788 

114.2S 

3^267  5345 

118.85 

41 

..46 

9739 

106.07 

^185  S7?9 

:  10. cs 

.226 

2647 

114.36 

.268  2478 

118.93 

42 

.147 

6105 

106.14 

.186  5346 

1 10. 1 5 

.226 

95 II 

"4-43 

.268  9616 

1 1 9.01 

43 

.14>* 

2475 

106.20 

.187  1957 

I  10.22 

.227 

6379 

114.51 

.269  6759 

119.09 

44 

.148 

8849 

106.27 

.187  SsvL 

110.29 

.22S 

3252 

114.58 

.270  3907 

119.17 

45 

3'i49 

5227 

106.33 

3.188  5192 

110.36 

3.229 

0129 

114.65 

3.271  1060 

119.25 

40 

.150 

1609 

106.40 

.189  1815 

'110.43 

-•19 

7010 

114.73 

.271  8217 

119.33 

47 

.150 

7995 

106.46 

.189  8443 

110. c 

.230 

3896 

114.80 

■272  5  379 

119.41 

48 

.151 

43X5 

106.53 

.190  5075 

''■'■57 

.231 

0786 

114.88 

■273  2546 

119.49 

4!> 

.152 

0778 

106.59 

.191  1711 

110.64 

.231 

7681 

114.95 

.273  9717 

119.57 

50 

3.52 

7176 

106.66 

3.191  8351 

110.71 

3.232 

458' 

115.03 

3.274  6894 

119.65 

51 

•'53 

3577 

106.72 

.192  4906 

110.77 

•^33 

1484 

115.10 

■275  4^75 

11973  ' 

52 

•'53 

9983 

106.79 

•'93  '^44 

110.84 

.233 

8392 

115.17 

.276  1261 

119.81 

53 

•'54 

6392 

106.85 

•'93  *'-97 

110.91 

.234 

S3°5 

"5-25 

.276  8452 

119.89 

54 

•«55 

2805 

106.92 

•'94  4954 

110.98 

•235 

2222 

115.32 

.277  56.^7 

119.97 

55 

3^'55 

9222 

106.99 

3.195  1615 

111.05 

3^235 

9'44 

115.40 

3.278  2848 

120.05 

50 

.156  5643 

107.05 

.195  8281 

I  :  i.I2 

.236 

6070 

"5-!7 

.279  0053 

120.13 

57 

•157 

20O8 

107  12 

.196  4950 

111.19 

.237 

3001 

"5  55 

.279  7263 

120.21 

58 

•'57 

8497 

107.18 

.197  1624 

111.26 

.237 

9936 

115.62 

.280  4477 

120.29 

59 

.158 

4930 

107.25 

.197  8302 

"'•34 

.238 

6876 

115.70 

.:'.8i  1697 

120.37 

00 

3'59 

1367 

107.31 

3.198  4984 

111.41 

3^239 

3820 

"5^77 

3.281  8921 

120.45 

f03 


TABLE  VI. 

For  finding  the  True  Anoinnly  ny  t\-2  Ti.nc  Tium  the  i'orilielion  in  a  P:.rabolic  Orbit. 


V. 

152° 

«  "' 

163° 

154° 

155 

1 

lo). 

•M. 

1)1  ir.  1". 
120.45 

I"). 

M. 

Di/r.  I". 
125.46 

loB  M. 

iiifr.  1". 

lof. 

M. 

oiir.  1". 

O' 

3.181 

8921 

3.326 

I'^'^i 

3.372  2684 

130.85 

3.420 

4064 

115.66 

1 

.2X2 

6151 

120.53 

.326 

8978 

i25^55 

•373  o53'< 

130.94 

.421 

2266 

136.76 

a 

.2^ 

33>*5 

120.61 

.327 

6513 

125.63 

•373  *'397 

131.04 

.422 

0475 

136.86 

3 

.284 

0624 

120.69 

.328 

4054 

125.72 

.374  6262 

131. 13 

.422 

8690 

136.96 

4 

.284 

7868 

120.77 

.329 

1600 

125. Si 

•375  4'33 

131.22 

.423 

6910 

137.06 

5 

3.285 

51 16 

120.85 

3329 

9151 

125.89 

3.376  2009 

i3'-32 

3.424 

5'37 

137.16 

0 

.286 

2370 

120.93 

.330 

6707 

125.98 

.376  9890 

131.41 

.425 

3370 

137.26 

7 

.286 

9028 

121.01 

.331 

4268 

126.07 

•377  7778 

131.50 

.426 

1609 

'37-37 

8 

.287 

6891 

121.10 

.332 

1835 

126.16 

.378   5671 

131.60 

.426 

9854 
8105 

'37-47 

» 

.288 

4160 

121.18 

-33^ 

9407 

126.24 

•379  3570 

131.69 

•427 

'37-57 

i     10 

3.2S9 

H33 

121.26 

3-333 

6984 

126,33 

3.380   1474 

'3'-I2 

3.428 

6362 

137.67 

1     11 

.289 

8711 

121.34 

•334 

.567 

126.42 

.380  9384 

131.88 

•429 

4626 

'37-77 

i    la 

.290 

5993 

121.42 

-335 

2154 

126.51 

.381   7300 

131.98 

.430 

2895 

137.88 

!     13 

.291 

3281 

121.50 

■335 

9747 

126.59 

.382  5221 

132.07 

•43' 

1171 

137.98 

1     ** 

.2  )2 

0574 

121.59 

.336 

7346 

126.68 

•3«3  3'48 

132.16 

•43' 

9452 

138.08 

i     15 

3.292 

7872 

121.67 

3-337 

4949 

126.77 

3.384  1081 

132.26 

3-432 

7740 

138.18 

i     1« 

.293 

5'74 

121.75 

•33** 

2558 

126.86 

.384  9019 

132.35 

•433 

6034 

138.29 

17 

.294 

2481 

121.83 

-339 

0172 

126.95 

.385  6963 

'32-45 

-434 

4334 

'38-39 

18 

.294 

9794 

121.91 

-3  39 

7792 

127.03 

.386  4913 

'32-54 

-43  5 

2641 

138.49 

19 

.295 

7HI 

122.00 

.340 

54>7 

127.17. 

.387  2869 

132.64 

.436 

0953 

138.59 

20 

3.296 

4433 

122.08 

3-34' 

5?F 

127.21 

3.388  0830 

132.73 

3-436 

9272 

138.70 

i     21 

.297 

1761 

122.16 

•342 

0682 

127.30 

.388  8797 

132.83 

-43- 

7597 

138.80 

1   aa 

.297 

9093 

122.24 

.342 

8323 

127.39 

.389  6770 

132-93 

.438 

5928 

138.90 

1     83 

.298    6430 

122.33 

-343 

51(09 

127.48 

-39     4749 

133.02 

•439 

4266 

139.01 

1     24 

.299 

3772 

122.41 

-344 

3020 

I27^57 

-391   2733 

133.12 

.440 

2609 

•I  39.11 

25 

3.300 

1119 

122.49 

3-345 

1277 

127.66 

3.392  0723 

133.22 

3-44' 

0959 

139.22 

2« 

.300 

847  F 

122.58 

-345 

8939 

'27-75 

.392  8719 

'33-3' 

•44' 

93'5 

139.31 

27 

.301 

5828 

122.66 

.346 

6606 

127.84 

-393   6720 

'33-4' 

.442 

7677 

1 39  42 

28 

.302 

3190 

122.74 

■^H 

4279 

127.93 

•394  4728 

'33-50 

•443 

6046 

'39-53 

1     29 

•303 

0557 

122.83 

.348 

1958 

128.02 

•395   274' 

133.60 

•444 

4421 

'39-63 

30 

3-303 

7929 

122.91 

3.3.18 

9641 

128.11 

3.396  0760 

133.70 

3-445 

2802 

'  39-74 

31 

.304 

53q6 

122.99 

•349 

7330 

128.19 

.396  8785 

133-79 

-446 

1189 

139.84 

32 

.305 

268S 

123.08 

.350 

5024 

128.28 

.397  6815 

133-89 

•446 

9583 

'39  95 

!     33 

.306 

0075 

123.16 

•351 

2724 

12S.37 

.398  4852 

'33-99 

•447 

7983 

140.05 

34 

.306 

7468 

123.24 

-352 

0429 

128.46 

•399  2894 

134.09 

.448  6389 

140.16 

35 

3-307 

4865 

'23-33 

3-35i 

8140 

128.55 

3.400  0942 

134.19 

3-449 

4802 

140.26 

30 

.308 

2267 

123.41 

•353 

5856 

128.65 

.400  8996 

134.28 

.450 

3221 

140.37 

37 

.308 

9674 

123.50 

•354 

3577 

128.74 

.401   7056 

134.38 

•45' 

1646 

140.47 

38 

.309 

7086 

123.5& 

•355 

1304 

128.S3 

.402   5122 

134.48 

•452 

0077 

'4°  57 

39 

.310 

4504 

123.66 

•355 

9037 

128.92 

•403   3193 

'34-57 

•452 

851s 

140.68 

40 

3. 311 

1926 

113-75 

3-356  6774 

129.01 

3.404  1270 

134.67 

3453 

6959 

140.79 

41 

.311 

93  i4 

123.83 

•557 

45'7 

129.10 

-404  9354 

'34-77 

•454 

5410 

140.90 

42 

.312 

(1786 

123.92 

.358 

2266 

1 29, 1 9 

-405   7443 

134.87 

•455 

3867 

141.00 

43 

-313 

4:24 

124.00 

-359 

0020 

129.28 

.406   5538 

134-97 

.456 

2330 

141. II 

44 

-3'4 

1667 

124.09 

-359 

7780 

129.37 

.407   3639 

135.07 

•457 

0800 

141.21 

45 

3-314 

9115 

124.17 

3.360 

5545 

129.46 

3.408   1746 

135.16 

3-457 

9276 

141.32 

46 

-315 

6567 

124.26 

.361 

3316 

129.56 

.408  9859 

135.26 

.458 

7759 

141.43 

47 

.316 

4025 

124.34 

.362 

l°J- 

i:;.6  5 

•409  7977 

'35-36 

•459 

6248 

141.54 

48 

-3>7 

1489 

124.43 

.362 

8873 

129.74 

.410  6102 

135.46 

.460 

4743 

141.04 

49 

.317 

8957 

124.51 

-363 

6660 

129.83 

.411   4233 

135.56 

.4.61 

3245 

141.75 

50 

3.318 

6430 

124.60 

3364 

4453 

129.92 

3.412  2369 

135.66 

3.462 

'753 

141.86 

51 

-3'9 

3909 

124.68 

.365 

2151 

130.01 

.413   0512 

135.76 

.463 

0268 

141.97 

52 

.320 

1392 

124.77 

.366 

0055 

130.11 

.413   8660 

135.86 

.463 

87S9 

142-07   . 

53 

.320 

8881 

124.86 

.366 

7864 

130.20 

.414  6815 

'35^96 

.464 

73'7 
585' 

142.18 

54 

.321 

O375 

124.94 

.367  5679 

130.29 

■4>5  4975 

136.06 

.465 

142.29 

55 

3.322 

3!-',  4 

125.03 

3.368 

3499 

130.38 

3.416   3142 

136.16 

3.466 

4392 

142.40 

50 

.323 

1379 
8888 

125.11 

.369 

1325 

130.48 

.417   1314 

136.16 

.407 

2939 

141.51 

57 

•323 

125.20 

.369 

9156 

'3057 

.417  9^92 
.418   7677 

136.36 

.468 

1492 

142. bl 

58 

•314 

6403 

125.29 

.370 

6993 

130.66 

136.46 

•469 

0051 

U2.'2 

59 

■325 

3923 

125.37 

•37« 

483b 

130.76 

419   5S67 

136.56 

.469 

86it> 

.;...8l 

00 

3.326 

1448 

125.46 

3-37* 

2684 

130.85 

3.420  4064 

136.66 

3^47- 

7192 

1 1.'  -  ■ 

6U1 


ibolic  Orbit. 


155 

0 

M. 

oiir.  1". 

4064 

115.66 

2266 

136.76 

0475  I 

136.S6 

8690  i 

136.96 

6910 

137.06 

5«37 

137.16 

3370  i 

137.16 

1609 

«37-37 

9«S4 
8105 

137-47 

137-57 

6362 
4626 

137.67 

»37-77 

i«95 

137.88 

1171 

137.9*! 

9452 

138.0H 

774° 

138.18 

6034 

138.29 

4334 

138.39 

1^8.  IQ 

i3«-59 
138.70 
138.80 
138.90 
139.01 
•139.11 

139.22 
139.32 
1 39  41 

139-53 
139.63 

>  39-74 
139.84 

139-95 
140.05 
140.16 

140.26 
140.37 
140.47 
14057 
140.68 

140.79 
140.90 
141.00 
141.11 
141.21 

141.31 

141.43 
141.5+ 
141.64 
141.75 

141.86 

141.97 
142.07 
142.18 
142.29 

142.40 
142.51 
142.61 


TABLE  VI. 

for  finding  the  True  Anomaly  or  tlie  Time  from  tlie  Perihelion  in  a  Parabolic  Orbit. 


I 

•J  ! 


7192         I|.T 


V. 


o 
1 

2 
3 
4 

5 
G 

7 
8 
0 

10 
11 

vz 

13 
14 

15 
14( 
It 


•Zi 
TZ 
23 
24 

25 
20 

27 
28 
29 

30 
31 
32 
33 
31 

35 
30 

37 
38 
30 

^o 
41 
43 
43 
44 

45 
40 
47 

48 
40 

50 
51 
52 
53 
54 

55 
50 
57 

58 
59 

60 


156^ 


log  M.         I 

I 

470  7192    i 

471  5772 

472  4358 

473  ^95" 

474  JSSO 

475  o'S6 

475  8769 

476  7388 

477  6014 

478  4646 

479  3185 

480  1931 

481  0583 

481  9242 

482  7907 

483  6579 

484  5258 

4S5   3944 
i86  2636 

■'■''7   133s 

488  0040 
4S8  8752 

489  747a 

490  6198 

491  4930 

492  3670 

493  i4>6 

494  1168 

494  9928 

495  8695 

496  7468 

497  6248 

498  S°3S 

499  3828 

500  2629 

501  1436 

502  02,0 

502  90 !■ 

503  7''-.>.; 
SCI  '  '34 

}''\    5o''' 

507  320- 

508  2143 

509  1012 

509  9889 

510  8772 

511  7662 

512  6560 

513  5464 

514  4375 

515  3294 

516  2219 

1    !  I  <,  I 

■■;'■:  cc9r 

3-5''^   N- 
,519  r,o 
.51c  6951 
.52?  5918 
.52i  4893 

3-5-3  3875 


Diff.  1". 


»4349 
143.60 

>43-7i 
143.82 

»43-93 

144.04 

't4-«5 
144.26 

'44-37 
144.48 

144-59 
144.70 

144-81 

144-93 
145.04 

'45-15 
145.26 

'45-37 

'45-49 
145.60 

145-71 
145.82 

'45.94 
146.05 

146.16 

146.28 
146.39 
146.50 
146.62 
146.73 

146.85 
146.96 
1+7.08 
JA7.19 
'  -/-31 
•.•.-.42 

-t'  .-^4 
J. 47.1-5 

'  -  7 
.^■,.^» 

148.00 
148.11 
148.23 
148.34 
148.46 

148.58 
148.70 
148.81 
148.93 
149.05 

'4;- '7 
149.28 
149.40 

149-51 
149.64 

'49-/5 


157= 


loK  M. 


3875 
2864 

i860 
0863 

98:3 

88  JO 

79'5 
6947 

59\5 

5031 

4085 

3 '45 
2213 
1288 
0370 

9459 
8556 
7660 
6771 
5890 

5015 
4148 
3289 
2430 
1591 

0754 
9924 

9101 
8285 
7477 
550  6677 

55'  5883 

552  5097 

553  43'9 

554  3548 

2785 
2029 
i.'.8o 


523 
5^4 
525 
526 
526 

527 
528 
529 
530 
53' 

532 
533 
534 
535 
536 

536 

537 
538 

539 
540 

541. 
542 

5^3 
544 
545 

546 
546 

547 
548 

549 


555 
556 
557 
558 
558 

559 
560 
561 
562 
5''3 
564 

565 
566 
567 
568 

569 

570 
571 
572 
572 

573 
574 
57? 
.76 
577 

3.578 


0539 
9806 

9080 
8361 
7650 
6947 
C-251 

5562 
4882 
4209 

3543 
2885 

2235 
1592 

°9:7 
0330 
9710 

9098 
8494 

7897 
7308 
6727 

6154 


Diff.  1". 

49-75 
49.87 

j.9.99 
50.1 1 

50.23 

50-35 
50.47 

50.59 

50.71 
50.83 

5°-95 
51.07 
51.19 
51.31 
5'-43 

5'-S5 
51.67 

5'-79 
51.91 
52.04 

52.16 
52.28 
52.40 

5252 
52.65 

52.77 
52.89 
53.01 

53-'4 
53.26 

53-38 
53-5' 
53-63 
53-75 
53-88 

54.00 
54- '3 
54-25 
54.38 
54.50 

54.63 

54-75 
54-88 
55.01 

55-'3 

,-6 

s-3^- 
5:1 

55-  •+ 
55-7^) 

55.89 
56.02 

56.15 
56.27 
56.40 

56-53 
56.66 
56.79 
56.92 
57.04 

'57-'7 
60.^ 


158= 


loB  M. 


3-578 

-579 
.580 
.581 
.582 

3-583 
-584 
.585 
.586 
.587 

3.588 

-589 
.589 
.590 
•59' 
3.592 

•593 
-594 
•595 
•596 

3-597 
•598 

•599 

.600 
.601 

3.602 
.603 
.6-- 1 
.605 
.606 

3.607 
.608 
.609 
.610 
.6ti 

3.612 
.613 
.614 

.615 

3.616 
.617 
.618 
.619 
.620 

3.621 
.622 
.623 
.624 
.625 

3.626 
.627 
.628 
.629 
.630 

3^63' 
.632 

•633 
•634 
-635 


Diff.  1". 

57-'7 
57^30 
57^43 
57-56 
57.69 

57.82 

57-95 
58.08 
58.21 
58-34 
58.47 
58.61 

58.74 
58.87 
59.00 

59-'3 
59.26 
59.40 

59-5  3 
59.66 

59-79 
59-93 
60.06 
60.19 
60.33 

60.46 
60.60 
60.73 
60.87 
61.00 

61.14 
61.27 
61.41 
61.54 
61.68 

61.81 
61.95 
62.09 
61.22 
62.36 

62.50 
62.63 
62.77 
62.91 
63.05 

6^.18 
63-32 
63-46 
63.60 

«-'37-i- 

63.88 
64.02 
64.16 
64.30 
64-44 
64.58 
64.72 
64.86 
65.00 
65.14 

3.636  6351       165.28 


6154 

5588 
5030 
4480 
3937 

34°3 
2876 

2357 
1846 

1342 
0847 

0359 
9880 
9408 
8944 

8488  ; 
8040  } 
7600 
7167 
6743 

6327  I 

59'9 

5518 

5126 

4742 

4365 
3997  I 
3637  i 
3285 
2941  j 

2605 

2277 

'957 
1646 
1342 

1047 
0760 
04S1 
0210 
9948 

9693 
9447 
9209 
8980 
8758 

8545 
8340 
8.44 

795" 
7776 

7604 

744' 
7287 
7140 
7002 

6873 
6751 
6638 

6534 
6438 


159= 


log  M. 


3-636 

•637 
.638 

•639 
.640 

3.641 
.642 

-643 
.644 

-645 

3-646 
.647 
.648 
.649 
.650 

3.651 
.652 

•653 
.654 

•655 

3.656 
.657 
.658 

•6j9 
.650 

3.661 
.662 
.663 
.664 
.665 

3.666 
.667 
.668 
.669 
.670 

3.671 
.672 
.673 
.674 

.675 

3.676 
.678 

-679 
.680 
.68: 

3.68  , 
.68  ; 
.68, 
.68; 
'^26 

1.687 
.688 
.689 
.690 
.691 

1.692 
.693 

•694 
.695 
.696 


635' 
6272 
6202 
6140 
6087 

6042 
6006 
5978 
5959 
5948 

5946 

595  3 
5968 

5992 
6025 

6066 
6116 

6175 
6242 
6318 

6403 
6497  I 

6599  ! 
6710  j 
6830 

6959  1 
7096  I 

7243  i 
7398  I 
7562  I 

7735  i 
79'7  i 
8108  I 
8308  1 
8516  i 

8734  I 
8961  I 
9196  I 

9441  ': 
9694  j 

9957 
0228  I 
0509 
0799 
1098 

1406 
1723 

20.)  9  , 
2384  , 
2728 

3082  I 

3445  I 

38,7  ; 

4' 98 
4588  j 

4988  i 
5397  ; 
5815 
624? 
6680 


3.697  7126 


Diff.  1". 

65.28 
65.42 
65.56 
65.71 
65.85 

65-99 
66.13 
66.28 
66.42 
66.56 

66.71 
66.85 
66.99 
67.14 
67.28 

67.42 

67-57 
67-72 
67.86 
68.01 

68.15 
68.30 
68.45 
68.59 
68.74 

68.89 
69.03 
69.18 
60,33 
6^.48 

69.62 
69.77 
69.92 

70.07 
70.22 

70.37 
70.52 
70.67 
70.82 
70.97 

71.12 
71.27 
71.42 

7'^57 
71.72 

71.87 
72.03 
72.18 

72.33 
72.48 

72.64 

72.79 

72-94 
73.10 

73-25 
73.40 
73-56 
73-7' 
73-87 
74.02 

74.18 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  tlie  Perihelion  in  a  Parabolic  Orbit. 


i 


V. 

0' 

160° 

•      161 

0 

162 

0 

163° 

log  M. 

Diff.  1". 

log  M. 

Di(T.  1". 

log  M. 

Diir.  1". 

logM. 

Diflf.  1".    \ 

207.00  ; 

5.697    7126 

174.18 

3.762    1539 

183.99 

5.830   3147 

194.87 

3.902   6107 

1 

.698    7581 

»74-34 

.763    2584 

184.16 

.831    4845 

195.06 

•9°3  8534 

207.21   ' 

1       '2 

.699    8046 

174.49 

.764    3639 

184.34 

.832   6554 

195.25 

.905  0973 

207.43 

3 

.700   8520 

174.65 

.765    4704 

184.51 

.833    8275 

'95^44 

.906  3425 

207.64 

4 

.701    9003 

174.S0 

.766    5780 

184.68 

.835    0008 

195.64 

.907  5890 

207.86 

5 

5.702    9496 

174.96 

3.767    6867 

184.86 

3.836    1752 

195.83 

3.908  8368 

208.08 

G 

.703    9999 

175.12 

.768    7963 

185.03 

.837    3508 

19G.02 

.910  0859 

208.29 

7 

.70s    05  1 1 

175.28 

.769   9070 

185.20 

.838    5275 

196.22 

.911   3363 

208.51 

8 

.706    1032 

'75^43 

.771    0187 

185.38 

•839   7054 

196.41 

.9.2  5880 

208.72 

9 

.707    1562 

I75^59 

.772    1315 

185-55 

.840   8844 

196.60 

.913  8410 

208.94 

10 

5.708    2102 

i75^75 

3^773  2454 

185-73 

[.842    0646 

196.80 

3.915  0953 

209.16 

11 

.709   2652 

175.91 

•774  3603 

185.90 

.843   2460 

196.99 

.916  3509 

209.38 

12 

.710    321I 

176.07 

•775  4762 

186.08 

.844  4286 

197.19 

.917  6078 

209.60 

13 

.711    3780 

176.22 

V6   5932 

186.25 

.845   6123 

197.38 

.918  8661 

2og.8i 

11 

.712  4358 

176.38 

•    ■■,      "712 

186.43 

.846   7972 

197.58 

.920  1256 

210.03 

15 

5.713    4946 

176.54 

3-V/' 

186.60 

5.847   9833 

197.78 

3.921  3865 

■iio  •».; 

16 

•7»4  5543 

176.70 

.779 

186.78 

•849    '705 

'97-97 

.922  6487 

210.48 

17 

.715  6150 

176.86 

.781    c 

186.96 

.850   3589 

.98.17 

.923  9122 

210.70 

18 

.716  6766 

177.02 

.782  1940 

.87.14 

.851    5486 

198.37 

.925   1770 

210.92 

10 

.717  7392 

177.18 

•783  3»74 

.87.3, 

.852   7394 

198.57 

.926  4432 

211.14 

20 

3.718  8028 

I77^34 

3.784  4418 

187.^9 
187.67 

5.853    9314 

198.76 

3.927  7107 

211.36 

21 

.719  8673 

177.50 

.785  5672 

•855   «245 

198.96 

.928  9795 

211.58 

22 

.720  9328 

177.66 

.786  6938 

'^Z-^5 

.856   3189 

199.16 

.930  2497 

211.81 

23 

•7i«   9993  . 

177.83 

.787  8214 

188.03 

-857  5»45 

199.36 

.931   5212 

212.03 

24 

.723  0668 

178.00 

.788  9501 

J88.21 

.858    7»12 

199.56 

.932  7940 

212.25 

25 

3.724  1352 

178.15 

3.790  0799 

188.39 

3.859  9092 

199.76 

3.934  0682 

212.48 

20 

.725   2045 

178.3. 

.791   2108 

188.57 

.861   1084 

199.96 

•935  3438 
.936  6207 

212.70 

27 

.726  2749 

178.47 

.792  3427 

188.75 

.862  3087 

200.16 

21293 

28 

.727   3462 

178.63 

•793  4757 

188.93 

.863  5103 

200.36 

•937  8989 

213.15 

29 

.728  4185 

178.80 

.794  6098 

189.11 

.864  7131 

200.56 

•939  '785 

213.38 

30 

3.729  4918 

178.96 

3-795   7450 
.796   8812 

189.29 

3.865  9171 

200.77 

3.940  459c 

213.61 

31 

.730  5661 

179.13 

189.47 

.867   1223 

200.97 

•94'   74'8 

213.83 

32 

.731   6413 

179.29 

.798  0186 

189.65 

.868   3287 

201.17 

•943  0254 

214.06 

33 

.732  7176 

>79^45 

•799   '571 

189.83 

.869  5363 

201.37 

944  3'05 

214.29 

34 

•733  7948 

179.62 

.800  2966 

100.01 

.870  7452 

201.58 

•945   5969 

214.52 

35 

5.734  8730 

179.78 

3.801   4372 

l'^O.20 

3.871   9552 

201.78 

3.946  8847 

214.74  ■ 

36 

•735   95ii 

'79^95 

.802  5790 

190.38 

.873    1665 

201.98 

.948   1738 

214-97   . 

37 

•737  0324 

180.11 

.803   7218 

190.56 

•874  379' 

202.19 

•949  4644 

215.20 

38 

.738   1136 

180.28 

.804  8657 

190.65 

.875   5928 

202.39 

.950  7563 
.952  0496 

215.43 

39 

•739  >957 

180.45 

.806  0108 

190.93 

.876  8078 

202.60 

215.66 

40 

3.740  2789 

180.61 

3.807   1569 

191.11 

3.878  0240 

202.80 

3^953  3443 

216.90 

41 

.741  3631 

180.78 

.808   3041 

191.30 

.879    2iI4 
.i<80    4601 

203.01 

•954  6403 
•955  93/8 

216.13 

42 

.742  4482 

180.94 

.809  4525 

191.48 

203.22 

216.36   • 

43 

•743   5344 

181.11 

.810  6020 

191.67 

.881     6800 

203.42 

■957  2366 

216.59 

44 

.744  6216 

181.28 

.811   7525 

191.86 

.882    9012 

203.63 

.958  5369 

216.82 

45 

3.745  7097 

181.45 

3.S12  9042 

192.04 

5.884    1236 

203.84 

3.959  8385 

217.06 

40 

.746  7989 

181.61 

.814  0570 

192.23 

•885   3473 

204.05 

.961   1416 

217.29 

47 

■747  8891 

181.78 

.815  2110 

I92.AI 
192.60 

.886  5722 

204.26 

.962  4460 

217.5^ 
217.76 

48 

.748  9803 

181.95 

.816  3660 

•887  7983 

204.46 
204.67 

.963  7519 

49 

.750  0725 

182.12 

.817   5222 

192.79 

.889  0257 

.965  0592 

218.00 

50 

3.751    1657 

182.29 

3.818  6795 

192. 9S 

3.890  2544 

204.88 

3.966  3678 

21V-3   1 

51 

•752   ^599 

182.46 
182.63 

.819  8379 

193.16 

.891  4843 

205.09 

.967  6779 

218.47 

52 

•753   3552 

.820  9974 

«9335 

.892  7155 

205.31 

.968  9895 

218.70  ; 

53 

•754  45H 

182.80 

.822   1581 

«93-54 

.893  9480 

205.52 

.970    t024 

.971  6168 

218.9+ 

54 

•755  5487 

182.97 

•823  3«99 

193.73 

.895  1817 

205.73 

219.18 

55 

5.756  6470 

183.14 

3.824  4829 

'93-9* 

3.896  4167 

205.94 

3.972  9326 

219.66 

56 

•757  7464 

183.31 

.825  6470 

194.11 

.897  6529 
.898  8905 

206.15 

•974  2498 
-975  5684 

57 

.758  8467 

183.48 
183.^5 

.826  8122 

>94-3° 

206.36 

219.90 

58 

•759  948' 

•827  9785 

194.49 
194.68 

.900  1293 

206.57 

.976  8885 

220.13 

50 

.761  0505 

183.82 

.829  1460 

.901   3694 

206.79 

.978  2100 

220.37 

GO 

3.762  1539 

183.99 

3.830  3147 

194.87 

3.902  6107 

207.00 

3-979  5330 

220.61 

606 

)olic  Orbit. 


TABLE  VI. 

For  fincUng  the  True  Anomaly  or  tlie  Time  from  the  Pcriiielion  in  a  Parabolic  Orbit. 


163^ 


M. 

5io7 
8534 
0973  i 

5890  ! 

8368 
0859 

33''3 
5880 
8410 

0953 
3509 
6078 
8661 
1256 

3865 
6487 
9122 
1770 
4432 


Diff.  1". 

207.00 
207.21 
207.45 
207.(14 
207.86 

208.08 
208.29 
208.^1 
1  208.72 
208.94 

209.16 
209.38 
209.60 
209.81 

210.03 

210.48 
210.70 
210.92 
211. 14 


7107 

9795 
2497 
5212 

794° 

0682 

3438 

6207 

7  8989 

)  1785 


211.36 
211.58 
211. 81 
212.03 
212.25 

212.48 


4595 
7418 
0254 
3105 
5969 

8847 
.8  1738 
9  4644 

;o  7563 

;2  0496 

13  3443 

14  6403 

57  2366  1 

58  5369 

59  8385 

61  1416 

62  4460 

63  7519 

65  0592 

66  3678 

67  6779 

68  9895 
70  J024 
171  6168 

,71  9326 
174  2198 

»75  5684 
,76  8885 
(78  2100 

>79  533° 


212.70 

21293 

213.15 

2I3.3S 

213.61 

213.83 

214.06 

214.19 

j  214-5^  : 

114.74 

114-97 
215.20 

215.43 
215.66 

216.90 

216.13        ; 

216.36       * 

216.59 

216.82 

217.06 
217.29 

217-5? 
217.76 
218.00 

218^23 
218.47 
218.70 
218.94 
219.18 


0- 

164'' 

165° 

166° 

167°        1 

1 

I"(- 

M. 

Diff.  1". 

log 

M. 

Diff.  1". 

log 

M. 

Diff.  1". 

lOB  M. 

Diff.  1". 

3-979 

533° 

220.62 

4.061 

6673 

236.01 

4.149 

7198 

253-57 

4.244 

5537 

273.78 

1 

.9X0 

8574 

220.86 

.063 

0842 

236.28 

.151 

2422 

253.88 

.246 

1975 

274.14 

•i 

.982 

1833 

221.10 

.064 

5027 

236.56 

.152 

7664 

254.19 

.247 

8434 

274.51  I 

3 

•983 

5106 

221.34 

.065 

9229 

236.83 

.154 

2925 

254.51 

.249 

4916 

274.87  1 

4 

.984 

8394 

221.58 

.067 

3447 

237.11 

-'55 

8205 

254.83 

.251 

1419 

275-24  1 

5 

3.986 

1696 

221.83 

4.068 

76S2 

237-39 

4- '57 

35°4 

25S-'4 
255.46 

4.252 

7944 

275.60  ! 

» 

.987 

5013 

222.07 

.070 

'933 

237.66 

.158 

8822 

.25A 
.256 

4491 

275-97  ' 

7 

.9S8 

8345 

222.31 

.07? 

6201 

237.94 

.160 

4159 

255.78 

1061 

276.34 

8 

.990 

1691 

222.56 

.073 

0486 

238.22 

.161 

9515 

256.10 

•257 

7652 

276.71 

0 

.991 

5051 

222.80 

.074  4787 

238.50 

.163  4891 

256.42 

.259 

4266 

277.08 

10 

3.992 

8427 

223.05 

4.075 

9106 

238.78 

4.165 

0285 

256.74 

4.261 

0902 

277-45  i 

11 

•994 

1817 

223.29 

.077 

3441 

239.06 

.166 

5699 

257.06 

.262 

7560 

277.82  ' 

vz 

•995 

5222 

223.54 

.078 

7792 

239-34 

.168 

1132 

257-38 

.264  4240 

278.20  1 

13 

.996 

86j-, 

223.79 

.080 

2161 

239  fl2 

.169 

6585 

257.70 

.266 

0943 

278.57  1 

11 

•998 

2077 

224.03 

.081 

6546 

239.^0 

.171 

2056 

258.02 

.267  7669 

278.95 

15 

3-999 

55^7 

224.28 

4.083 

0948 

240.18 

4.172 

7547 

258.35 

4.269 

44' 7 

279-32 

10 

4.000 

8991 

224.53 

.084 

5368 

240.46 

-'74 

3058 

258.67 

.271 

1187 

279.70 

17 

.002 

2471 

224.78 

.085 

9804 

240.75 

-'75 

85S8 

259.00 

.272 

7981 

280.08  1 

18 

.003 

5965 

225.03 

.087 

4257 

241.03 

-'77 

4138 

259-33 

•274  4797 

280.46  . 

19 

.004  9474 

225.28 

.088 

8728 

241.32 

.178 

9707 

259.65 

.276 

1635 

280.84  ' 

20 

4,006 

2999 

225.53 

4.090 

3215 

241.60 

4.180 

5296 

259-98 

4^277 

8497 

281.22  1 

21 

.007 

6538 

225.78 

.091 

7720 

241.89 

.182 

0905 

260.31 
260.64 

.279 

538' 

281.60  ! 

22 

.C09 

C093 

226.04 

•093 

2242 

242.08 

..83 

6534 

.281 

?''^i) 

281.98  '■ 

1  23 

.010 

3663 

226.29 

.094 

6781 

242. i;6 

.185 

2182 

260.97 

.282 

9219 

282.36  : 

1  2-1 

.011 

7248 

226.54 

.096 

1337 

242.75 

.186 

7850 

261.30 

.284 

6173 

282.75  : 

!  25 

4.013 

0848 
44^3 
8093 

226.79 

4.C97 

59" 

243.04 

4.188 

3538 

261.63 

4.286 

3 '49 

283.14  ' 

20 

.014 

227.05 

.099 

0502 

24333 

.189 

9246 

261.96 

.288 

0149 

283.52  ■ 

27 

.015 

227.30 

.100 

5110 

243.62 

.191 

4974 

262.30 

.289 

7172 

283.91 

28 

.017 

'739 

227.55 

.101 

9736 

2.-.3.91 

-'93 

0722 

262.63 

.291 

4218 

2S4.30 

20 

.018 

S400 

227.81 

.103 

4379 

X^4  20 

•'94 

6490 

262.97 

•293 

1288 

284.69 

30 

4.019 

9077 

228.06 

4.104 

9040 

244.4. 

4.196 

2278 

263.30 

4.294 
.296 

8381 

285.08 

31 

.021 

2769 

228.32 

.106 

37'8 

244.78 

•'97 

8086 

263.64 

5498 

285.47 

32 

.022 

6476 

228.58 

.107 

8414 

245.08 

.199 

39' 5 

263.98 

.298 

2638 

285.87 

33 

.024 

0199 

228.84 

.109 

3127 

245-37 

.200 

9764 

264.32 

.299 

9802 

286.26 

31 

.025 

3937 

229.09 

.110 

7858 

245.67 

.202 

5633 

264.66 

•3^- 

6990 

286.66  ; 

35 

4.026 

7691 

229^35 

4.112 

2607 

245.96 

4.204 

1523 

265.00 

4^3='3 

4201 

287.05  ! 

36 

.028 

1460 

229.62 

.113 

7374 

246.26 

.201; 

743  5 
3363 

lint 

.305 

1436 

287.45 

37 

.029 

5245 

229.88 

.115 

2158 

246.55 

.207 

.306 

8695 

287.85 

38 

.030 

9045 

230.14 

.116 

6960 

246.85 

.208 

93'4 

266.02 

.308 

5978 

288.25 

39 

.032 

2861 

230.40 

.u8 

1780 

247.15 

.210 

5286 

266.37 

.310 

3285 

288.65 

40 

4.033 

6693 

230.66 

4.119 

6618 

247^45 

4.212 

1278 

266.71 

4.312 

0616 

289.05 

41 

.035 

°540 

230.92 

.121 

1+^t 

247^75 

.213 

7291 

267.06 

.313 

7971 

289.45  : 

42 

.036 

f^'^'*- 

231.18 

.122 

6348 

248.05 

.215 

3325 

267.40 

-3'5 

535° 

289. X6 

43 

•037 

8283 

231.45 

.124 

1239 

248.35 

.216 

9379 

267.75 

-3'"' 

2753 

290.26  1 

44 

.039 

2177 

231.71 

.125 

6149 

248.65 

.218 

5455 

268.10 

.319 

0181 

290.67  ; 

45 

4.040 

6088 

231.97 

4.127 

1077 

248.95 

4.220 

'55' 

268.44 

4.32c 

7633 

291.07  ' 

40 

.042 

0015 

232.24 

.128 

6023 

249.25 

.221 

7668 

268.79 

.322 

51 10 

291.48 

47 

.043 

3957 

237,51 

.,30 

0988 

249.56 

.223 

3806 

269.14 

.324 

261 1 

291.89  : 

48 

•044 
.046 

7915 
1890 

232-77 

.131 

5970 

249.86 

.226 

9965 

269.50 

.326 

°IE 

292.30 

49 

233-04 

•'33 

0971 

250.17 

6:46 

269.85 

-327 

7688 

292.71 

50 

4.047 

5880 

233.31 

4-134 
.136 

5990 

250.47 

4.228 

2347 

270.20 

4-329 

^l^J 

293.13 

51 

.048  9887 

233-57 

1028 

250.78 

.229 

8570 

270.55 

-33' 

2863 

293-54 

52 

.050 

3909 
7948 

233.84 

•'37 

6084 

251.08 

-23' 

4814 

270.91 

-333 

0487 

293-95 

53 

.051 

234.11 

•'39 

1158 

251.39 

-233 

1079 

271.27 

-334 
.336 

8137 

294-37 

54 

-053 

2003 

234-38 

.140 

6251 

251.70 

-234 

7366 

271.62 

5812 

294.79 

55 

4.054 

6074 

234.65 

4.142 

1362 

252.01 

4.236  3674 

271.98 

4^338 

3511 

295.20 

50 

.056 

01 0 1 

234-92 

.143 

6492 
1641 

252.32 

.238 

0003 

272.34 

•34° 

1236 

295.62 

57 

.057 

^384 

^35.19 

•'45 

252.63 

239 

6354 

272.70 

•34' 

^986 

296.04 

:  58 

.058 

235.46 

..46 

6808 

252.94 

.241 

2727 

273.06 

•343 

6762 

296.47 

296.89 

59 

.060 

2520 

235-73 

.148 

'994 

253-25 

.242 

9121 

273.42 

•345 

4562 

\   00 

• 

4.061 

6673 

236.01 

4.149 

7198 

253-57 

4.244 

5537 

273.78 

4-347 

2388 

297.31 

' 

» 

607 

TABLE  VI. 

For  finding  tlie  True  Anomiily  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orldt. 


V. 

i 

0' 

16J 

log  M. 
4.347   2388 

Diff.  1". 

,      169° 

170° 

171° 

log  M. 
4.459    1242 

Diff.  1". 

325.07 

log  M. 

Diff.  I". 

log 

M. 

DUT.  1".    , 

398.87 

297.31 

4.581    9445 

358^3' 

4-7'7 

9835 

1 

.349   0240 

^97^74 
298.16 

.461    0761 

32^57 
326.08 

.584  0962 

358.92 

.720 

3790 

399.62 

3 

•35°  i*'"? 

.463   031 1 

.586    2516 

359-53 

.722 

7790 

400.38 

3 

.352  6019 

298.59 

.46^  9891 
.466   9501 

326.59 

.588   4106 

360.15 

.725 

1835 

401.14 

4 

•354  3948 

299.02 

327.10 

•590  5734 

360.76 

•7*7 

59*6 

401.90 

5 

4.356   1902 

*99-45 

4.468   9142 

327.61 

4.592  7398 

361.38 

4-730 

0063 

402.66 

0 

•357  9««2 

299.88 

.470   8814 

328.12 

•594  9'oo 

362.00 

.732 

4*45 

403.43 

7 

•359  7888 

300.31 

.472   8517 

328.64 

•597  0838 

362.62 

-734 

8474 

404.19 

8 

.361   5919 

300.75 

•474  8250 

329.15 

•599  *6i5 
.601  4428 

5^^o5 

-737 

2749 

404.96 

0 

•363  3977 

301.18 

.476  8015 

329.67 

363.88 

-739 

7070 

405.74  • 

10 

4.365  2061 

301.62 

4.478  7811 

330.19 

4.603  6280 

364.50 

4.742 

1438 

406.52 

11 

.367  0171 

302.05 

.480  763; 

330.71 

.605  8169 

365-'4 

-744 

5852 

407.30 

13 

.368  8308 

302.49 

.482  7495 

331.23 

.608  0096 

365^77 

•747 

0314 

408.08 

13 

.370  6470 

302.93 

.484  7385 

33'-75 

.610  2061 

366.40 

•749 

4822 

408.87 

14 

.372  4659 

303.37 

.486  7306 

332.28 

.612  4064 

367.04 

•75' 

9378 

409.66 

15 

4-37'4  2875 
.376   1117 

303.81 

4.488  7258 

332.81 

4.614  6106 

367.68 

4^754 

3981 

410.45 

16 

304.26 

.490  7242 

333^33 

.616  8186 

368.32 

.756  X632 

411.24 

17 

•377  93*^6 

3O4^70 

.492  7258 

333.86 

.619  0304 

368.96 

•759 

3330 

412.04 

18 

•379  7681 

305.15 

-494  7306 

334.40 

.621  2461 
•623  4657 

369.61 

.761 

8077 

412.84 

10 

.381   6003 

305-59 

.496  7386 

334-93 

370.26 

•764 

2872 

413.65 

20 

4-3»3  435^ 

306.04 

4.498  7498 

335-46 

4.625  6892 

370.91 

4.766 

7715 

414.46 

21 

.385  2728 

306.49 

.500  7642 

336.00 

.627  9166 

37'-56 

.769 

2606 

415.27 

'    22 

.387  1131 

306.94 

.502  7818 

336-54 

.630  1480 

372.21 

•771 

7547 

416.08 

23 

.388  9561 

307-39 

.504  8026 

33708 

.632  3832 

372.87 

•774 

*536 

416.90 

34 

.390  8019 

307^85 

.506  8267 

337.62 

.634  6224 

373^53 

.776 

7574 

417.72 

35 

4.392  '650? 

308.30 

4.508  8541 

338.16 

4.636  8656 

374-19 

4-779 

2662 

418.54 

!    26 

•394  5015 

308.76 

.510  8847 

338.71 

.639  1127 

374.86 

•7^ 

7799 

4'9-37 

37 

•39<>   3554 

309.21 

.512  9186 

339.26 

.641    3639 

37552 

.784 

2986 

4.70.20 

28 

.398   2121 

309.67 

.514  9558 

339.80 

.643  6190 

376.19 

.786 

8222 

421.03 

20 

.400  0715 

310.13 

.516  9962 

340.35 

.645  8781 

376.86 

.789 

3509 

421.86 

30 

4.401  9337 

310.59 

4.519  0400 

340.91 

4.648  1413 

377^53 

4-791 

8846 

422.70 

31 

.403  7986 

311.06 

.521  0871 

341.46 

.650  4085 

378.21 

-794 

4*33 

4*  3- 54 

32 

.405  6663 

311.52 

.523      376 

342.02 

.652  6798 

378.89 

.796 

9671 

424,39 

33 

.407  5368 

311.99 

.525   1913 

342^57 

•654  9552 

379^57 

-799 

5160 

42.^24 

34 

.409  4102 

312.45 

.527  2484 

343- '3 

.657  2346 

380.25 

.802 

0700 

4^.6.09 

1    35 

4. 411   2863 

312.92 

4.529  3089 

34369 

4.659  5182 

380.93 

4.804 

6291 

426.95 

1    36 

.413   1652 

3«339 

•531   37*8 

344.26 

.661   8059 

381.62 

.807 

'934 

427-81   , 

37 

.415  0469 

313.86 

•5  33  4400 

344.82 

.66a  0977 
.666  3936 

382.31 

.809 

762S 

42S.67  , 

38 

.416  9315 

3>4^33 

•535   5«o6 

345-39 

383.00 

.812 

3374 

4*9-53 

39 

.418  8189 

314.80 

•5  37  5846 

345-95 

.668  6937 

383.70 

.814 

9172 

430.40  , 

40 

4.420  7091 

315^28 

4.539  6620 

346.52 

4.670  9980 

384.39 

4.817 

5022 

431.28 

41 

.422  6022 

3'5-7S 

•54'   7429 

347-09 

.673  3064 

^^'•S^ 

.820 

?2P 

45*-'5 

42 

.424  4982 

316.23 

•543  '^^7^ 

347-67 

.675  6191 

385.80 

.822 

6881 

433-°3 

43 

.426  3970 

316.71 

•545  9149 

348.24 

.677  9360 

386.50 

.825 

2889 

433-91 

44 

.428  2987 

317.19 

.548  0061 

348.82 

.680  2571 

387.21 

.827 

8950 

434.80 

45 

4.430  2037 

317.67 

4.550  1007 

349.40 

4.682  5825 

387.92 

4.830 

5065 

435-69 

40 

.432  1108 

318.16 

.552  1989 

349.98 

.684  9121 

388.63 

.833 

1234 

436.59 

47 

.434  0212 

31S.64 

•554  3005 

350-56 

.687  2460 

389-34 

.835 

7456 

^n-^l 

48 

•435  9345 

3«9-»3 

.556  4056 

3S'-'5 

.689  5842 

390.06 

.838 

373* 

438.38 

40 

j 

•437  8507 

319.61 

•558  5'43 

3S'-73 

.691  9268 

390.78 

.841 

0062 

439.29 

1    50 

4.439  7698 

320.10 

4.560  6264 

351-3* 

4-694  2736 
.696  6248 

391.50 

4.843 
.846 

6446 

440.20 

51 

.441   6919 

3^0-59 

.562    7i2I 

.564  8614 

352.91 

39»^23 

2886 

441. II 

52 

.443  6169 

321.08 

35350 

.698  9803 

392.96 

.848 

9380 

442.03 

53 

•445  5449 

321.58 

.566  9842 

354.10 

.701   3402 

393.68 

.85, 

59*9 

44*-95    f 

54 

•447  4758 

322.07 

.569  1 106 

354-69 

.703  7046 

394.42 

.854 

*533 

443-!*7    1 

55 

4.449  4097 

3"-S7 

4.571   2405 

355-*9 

4.706  0733 

395^'5 

4.856 

9193 

444-8° 

56 

•451   3466 

323.06 

•573  3741 

355-89 

.708  4464 

395.89 

.859 

5909 

^*)il 

57 

•453  2865 

3*3-56 

•575  S"3 

356-49 

.710  8240 
.713  2060 

396.63 
397-38 

.862 

2680 

446.66 

58 

•455  "94 

324.06 

•577  6521 

357-10 

•!^+ 

9508 

447.60 

50 

•457  '753 

324.56 

•579  7965 

357.70 

•7'S  S9»S 

398.12 

.867 

6392 

■f48-54 

60 

4.459  1242 

325.07 

4.581  9445 

358.31 

4.717  9835 

398.87 

4.870 

3333 

449.49 

60S 


bolk-  Orl.it. 


171° 

M.        1 

1 

DUT.  1". 

9835 
379° 
7790 
1835 
5926 

398.87 
399.62 
400.38 
401.14 
401.90 

0063 

4245 
X474 
2749 
7070 

1438 
5852 

0314 

4822 

9378 
3981 

8632 

3330 

8077 

.  2872 

7715 

I  2606 

7547 

•  ^536 

1  7574 

(  266a 

;  7799 
|.  29S6 
)  8222 
)  3509 
t  8846 

^  4^33  t 
5  9671  ' 

)   5160  I 

2  0700  I 

4  6291 

7  1934  ! 
9  762S  I 
2  3374  i 

4  9«7a  I 

7  502*  ! 

o  0925 
2   6881  i 

5  2889 
7  895° 
,0  5065 
;3  1*34 
15  7456 
;8  373* 
[.I  0062 

^3  6446 
^6  2886 
1.8  9380 
;i  5929 
54  2533 

56  9193 

59  5909 
62  2680 
64  9508 
67  6392 

70  3333 


402.66 
403.43 
404.19 
404.96 

4°5-74 
406.52 
407.30 
408.08 
408.87 
409.66 

410.45 
411.24 
412.04 
412.84 
413.65 

414.46 

415.27 
416.08 
416.90 
417.72 

418.54 
1  4'9-37 

]     4.7  0.20 

1     421.03 

421.86 

422.70 

4*3-54 
4*439 
4*5-*4 
4-6.09 

426.95 
427.81 
428.67 
429.53 
430.40 

431.28 
452.15 

433-°3 
i  433-9I 

i  434'^° 

i  435.69 

1  436-59 

437-48 

438.38 

439.29 
440.10 
441.11 
442.03 
44*-95 
443-87 

444-80 

446.66 
447.60 
.H8-54 

449-49 


TABLE  VI. 

For  finding  tlit-  True  Anomaly  or  llie  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


tf. 


O' 
1 

u 

3 
4 

5 
6 

7 
8 
9 

10 
11 
12 
13 
14 

13 
IG 
17 

18 
19 

20 
21 
22 
23 
24 

25 
20 

27 
28 
29 

30 
31 
32 
33 
34 

35 
30 
37 
38 
39 

40 
41 
42 
43 
44 

45 

48 
47 

48 
49 

50 
51 
52 
53 
54 

55 
3G 
37 

58 
59 


172^ 


log  M. 


DilT.  1". 


4-870  3333 
.873  0331 

.875  7386 
.878  4499 
.881    1668 

4.883  8896 
.886  6182 
.889  3526 
.892  0929 
•894  8391 


4.897 

5912 

.900 

349* 

.903 

1132 

.905 

8831 

.90S 

6591 

4.911 

44" 

.914 

2291 

.917 

0233 

.919 

8235 

.922 

6290 

4-9*5 
.928 
.931 

•933 
.936 

4-939 
94* 
945 
948 

951 

4-953 
.956 

•959 
.962 
.965 

4.968 
971 
974 
977 
980 

4.983 
985 

991 
994 

4-997 

5.000 

.003 

.006 

.009 

5.012 
.015 
.018 
.021 
.024 

5.027 
.031 
.034 
.037 
.040 

5.043 


44*5 
2612 
0862 
9'74 
7549 

5987 
4489 
3053 
1682 

■^375 

9132 

7954 
6841 

5793 
481 1 

3S94 

3°44 
2260 

1543 
0893 


9795 
H8 
970 

8659 

8418 
8246 
8143 
Siii 
8148 

8256 
8435 
8685 
9006 
9399 
98O4 
0.J.02 
1013 
1697 
*454 
3285 


449.49 

450.44 

45'-39 

45*-35 

453^3i 

454.28 

455*5 
456.23 

457.20 

458.19 

459-«7 
460.16 
461.16 
462.16 
463.16 

464.17 
465.18 
466.20 
467.22 
468.25 

469.28 
47o^3' 
471-35 
47*-39 
473-44 

474-49 
475-55 
476.61 
477.68 

478.75 

479.83 
480.91 
481.99 
483.08 
484.18 

485.28 

486.38 

487.49 
488.61 

489.73 

490.85 
491.98 
493.12 
494,26 
495.40 

496.55 

497-71 
498.87 
500.04 
501.21 

502.39 

503-57 
504.76 

5°5-95 
507.15 

508.36 

509-57 
510.79 
512  01 
513.24 

514-47 


173^ 


los  M. 


Dlff.  1". 


043 
046 
049 
052 
055 

058 
061 
065 
068 

071 

074 
077 
080 
084 
087 

090 

093 
096 
100 
103 

106 
109 
113 
116 
119 


207 
211 
214 
218 
22  1 

225 
228 
232 
236 
239 

*43 


3285 
4191 

5171 
6226 

7356 
8562 

9843 

1202 
2637  I 
4149 

5738 
7406 
91 5 1 
0976 
2879 

4862 
6924 
9067  i 
1290  I 
3594  i 
5980  I 

8447  I 
0997  I 
3629  I 

6344  ! 


122  9143 
126  2026 
129  4992 
132  8044 
136  1181 

139  4403 
142  7711 
146  1106 
149  45S8 
152  8157 

156  1813 

159  5558 
162  9392 
166  3315 
169  7328 

173  1431 

176  5624 
179  990S 
183  4284 
186  8752 

190  3312 
193  7966 
197  2713 
200  7554 
204  2489 


7520 
2646 
7868 
3186 
8602 

4116 

97*7 
5437 
1247 
7156 

3165 


3» 


5 '4-47 
5'5-7' 
516.96 
518.21 
5 » 9-47 

520.73 
522.00 
523.28 
524.56 
525.85 

527.14 
528.44 

5*9-75 
531.06 
532.38 

533-71 
535-°4 
536.38 

537-73 
539.08 

540.44 
541.81 
543.18 
544-56 
545-95 

547-34 
548.74 
550.15 

551-57 
552.99 

554.42 
555-86 
557-30 
558.75 
560.21 

561.68 
563.16 
564.64 
566.13 
567.63 

569.13 

570.65 
572.17 
573-70 
575-*4 

576-78 
578-34 
579-90 
581.47 

583-05 
584-64 

587.84 

589-45 
591.07 

59*-7J 
59435 
596.00 
597-66 
599-3* 
601.00 

6U9 


174' 


lot?  M. 


*43 
246 

250 
*54 

257 

261 
265 
268 
272 
276 

279 
283 
287 
291 
294 

298 
302 
306 

309 
313 

317 
321 
3*5 
3*9 
33* 

336 

340 

344 
348 

35* 

356 
360 
364 
368 
37* 

376 
380 

384 
388 

39* 

396 

400 
404 
408 
412 

416 
420 

4*5 
4*9 
433 

437 
441 

445 
450 

454 

458 
462 

467 

47  > 
475 


3165 
9276 
5488 

1802 
8218 

4738 
1  361 
S089 
4922 
i860 

8904 
6055 

33'3 
0680 
8154 

5738 
343* 
1*57 
9152 

7179 

5319 

3571 
1938 
04 1 S 
9014 

7726 
6554 
5499 
4562 

3744 

3°45 
2466 
2007 
1 67 1 
1456 

1364 
1396 

1553 
1834 

2242 

2777 
3439 
4229 

5«49 
6199 

7379 
8692 
01  56 
1714 

34*7 

5*74 
7258 
9378 
1636 
4032 

6568 

9*44 
2062 
5022 
8125 


480   1373 


Diir.  1". 
601.00 

602. 6(; 
604.38 
606.08 
607.80 

609.53 
611.26 
613.00 

64-75 
616.52 

618.29 
620.08 
621.87 
623.67 
6*5-49 
627.31 
629.15 
631.00 
632.85 
634-72 

636.60 
638.49 
640.39 
642.30 
644.23 

646.16 
64S.1: 
650.07 
652.04 
654.02 

656,01 
658.02 
660.04 
662.07 
664.1 1 

666.17 
668.24 
670.32 
672.41 
674.52 

676.64 
678.77 
680  92 
683.08 
685.25 

687.44 
689.64 
691.85 
694.08 
696.33 

698.59 
700.86 
703.15 

705-45 
707.77 

710.10 
712.45 
714.81 
717.1; 
719-59 
722.00 


175 


loK  M. 

5-480    1373 

484  4765 
488  8304 

493  1989 
497  5823 

5.501  9806 

.506  3939 

.5  10  8223 

.515  •'.659 

.519  7248 


DifT.  1". 


5-5*4 
5*8 
533 
537 
54* 

5.546 

-55» 
-555 
.560 

•565 
5.569 
-574 
-579 
.583 
.58§ 

5-593 
•597 
.602 
.607 
.612 

5.617 
.621 
.626 
.631 
.636 

5.641 
.646 
.651 
.656 
.661 

5.666 
.671 
.676 
.681 
.686 

5.691 
.696 

.701 
.706 
.711 


1992 
6890 
1 946 
7158 
2529 

8060 

375' 
9605 
5621 
1S02 

8148 
4661 
1341 
8190 
5210 

2401 

9764 

7302 
5014 
2903 

0970 
9216 
7642 
625c 
5041 

4017 

3179 
2528 

2065 

1793 

1713 
1825 
21  32 
2635 
3336 

4236 

5337 
6640 
8147 
9860 


5.-^17  1779 

.722  390S 

.727  6247 

-73*  8798 

•738  1563 

5-743  4544 

.748  7742 

-754  «'59 
-759  4798 
-764  8659 

5.770  2745 


722.00 
724.42 
726.87 

7*9.33 
73  1.80 

734-3° 
736.81 

739-33 
741.87 

744-44 

747.02 
749.61 
752.23 
754.86 
757-5' 
760.18 
762.87 
765.58 
768.31 
771.05 

773-82 
776.61 

779-41 
782.24 
785.08 

787-95 
790.84 

793-75 
796.68 

799-63 
802.60 
805.60 
808.62 
811.66 
814.72 

817.81 
820.92 
824.05 
827.21 
830.39 

833.60 
836.83 
840.08 
843.36 
846.67 

850.00 

853-36 
856.75 
860.16 
863.60 

867.06 
870.56 
874.08 
877.63 
881.21 

884.82 
888.46 
892.13 
895.83 
899.56 

903.31 


TABLE  VI. 

For  finding  the  True  Anomnly  or  tlie  Tiiiu'  from  the  IVrilielion  in  a  Pariibolic  Orbit. 


1  " 

V. 

176° 

'      177 

0 

178° 

179° 

lo(i 

M. 

Dim  1". 

logM. 
6.144  62ji9 

Diff.  1". 
1205.3 

logM. 

DilT.  1". 

IokM. 

Diff.  1". 

0' 

?-770 

4745 

903.3 

6.672  5724 

1808.8 

7.575  4640 

3619 

1 

•775 

705X 

907,1 

.151  8807 

1212.0 

.68  3  4709 

1824.0 

•597  3596 

3680 

2 

.7S1 

"599 

910.9 

•'59  '733 

1218.8 

.694  4613 

1839.5 

.619  6295 

3744 

i   a 

.7S6 

6370 

914.8 

.166  5070 

1225.7 

-705  5454 

1855.3 

.642  2868 

3809 

'   4 

] 

.791 

"374 

918.7 

.173  8823 

1232.7 

.716  7248 

1871.3 

.665  3452 

3877 

« 

5-797 

661Z 

922.6 

6.181  2997 

1239.8 

6.728  0010 

1887.5 

7.688  8192 

3948 

0 

1      ^. 

.803 

2086 

926.6 

-'88  7597 

1246.9 

•739  3758 

1904.1 

.712  7239 

4021 

!   7 

.8oii 

7798 

930.6 

.196  2628 

I  254. 1 

.750  8509 

1921.0 

•737  0756 

4097 

8 

.814 

37S« 

934-6 

.203  8095 

1261.4 

.762  4279 

1938.2 

.761  8913 

4176 

» 

.819 

9946 

938.6 

.211  4002 

1268.8 

-774  '090 

'955-6 

.787  1889 

4257 

1  10 

5.825 

6386 

942.7 

6.219  0354 

1276.3 

6.785  8958 

'973-4 

7.812  9876 

4343 

11 

.83. 

3°73 

946.8 

.226  7158 

1283.8 

•797  7904 

1991.5 

.839  3075 

443' 

12 

.837 

0008 

951.0 

.234  4419 

1291.5 

•809  7946 

2010.0 

.866  1702 

4524 

1  Y'i 

.842 

7'95 

955-2 

.242  2142 

1299.2 

.821  9106 

2,028. g 

.893  5986 

4620 

i  14 

.848  4634 

959-5 

.250  0333 

1307.1 

•834  '404 

2048.0 

.921  6170 

4720 

15 

5.854 

2329 

963.7 
968.0 

6.257  8997 

1315.0 

6.846  4863 

2067.5 

7.950  2513 

4825 

10 

.860 

0282 

.265  8139 

1323.0 

.85S  9503 

20S7.3 

7.979  5292 

4935 

17 

.865 

8495 

972-4 

.273  7766 

'33'-' 

.871  5348 

2107.6 

8.009  4802 

5050 

18 

.871 

6970 

976.8 

.281  7884 

'339-4 

.884  2422 

2128.3 

.040  1361 

5170 

10 

.877 

5710 

981.2 

.289  8499 

'347-7 

.897  0749 

2149.4 

.071  5309 

5296 

20 

5.883 

47J7 

985.7 

6.297  9617 

1356.2 

6.910  0353 

2170.9 

8.103  701 1 

5428 

21 

.889 

3993 

990.2 

.306  1244 

1364.7 

.913  1 26 1 

2192.8 

.136  6857 

5568 

22 

.895 

3541 

994.8 

.314  3387 

'373^3 

,936  3498 

2215.2 

.170  5274 

5714 

23 

.901 

3365 

999.4 

.322  6052 

1382.1 

•949  7093 

2238.0 

.205  2717 

5869 

24 

1 

.907 

3465 

1004.0 

•33=  9247 

1391.0 

.963  2073 

2261.4 

.240  9679 

^032 

1  25 

5-913 

3845 

1008.7 

6.339  2977 

1400.0 

6.976  8466 

2285.2 

8.277  6700 

6204 

26 

.919 

4507 

1013.4 

•347  7249 

1409. 1 

6.990  6304 

2309.6 

.315  4361 

6387 

27 

.925 

5454 

1018.1 

.356  2072 

1418.3 

7.004  5616 

2334^3 

•354  3298 

6580 

j  28 

•93' 

6688 

1022.9 

.364  7451 

1427.11 

.018  6437 

2359-7 

•394  4205 

6786 

1  20 

-937 

8213 

1027.8 

•373  3395 

'437-' 

.032  8796 

2385.7 

•435  7842 

7004 

S  30 

5-9+4 

0030 

1032.7 

6.381  9910 

1446.7 

7.047  2729 
.061  8271 

2412.2 

8.478  5044 

7238 

31 

.950 

2144 
4550 

1037.6 

.390  7005 

1456.4 

2439.4 

.522  6731 

7488  ' 

1  32 

-95'' 

1042.6 

.399  46S7 

1466.2 

.076  5458 

2467.1 

.568  3920 

7755 

!  33 

.962 

7269 

1047.7 

.408  2965 

1476.2 

.091  4329 

2495.4 

.615  7739 

8042 

i  34 

.969 

0287 

1052.9 

.417  1846 

1486.4 

.106  4921 

2524.5 

.664  9442 

8352  . 

35 

S-975 

3613 

1058.0 

6.426  1337 

1496.7 

7.121  7276 

2554.2 

8.716  0431 
.769  2286 

8686  , 

36 

.981 

7249 

1063.2 

•435  '449 

1507.0 

•137  '434 

2584.6 

9048 

37 

.9S8 

1198 

1068.4 

•444  219' 

1517.6 

.152  7440 

2615.8 

.824  6779 

944' 

38 

5-994 

5464 

1073.7 

•453  3569 

1528.3 

.168  5336 

2647.6 

.882  5925 

9870  ; 

1  39 

6.001 

0050 

1079.1 

•462  5594 

1539.2 

.184  5171 

2680.4 

.943  2018 

10340  i 

j  40 

6.007 

4958 

1084.5 

6.471  8275 

1550.2 

7.200  6993 

2711.9 
2748.3 

9.006  7690 

10857  ; 

41 

.014 

0192 

1089.9 

.481  1620 

1561.3 

.217  0850 

•073  5974 

11429 

42 

.020 

5756 

1095.4 

.490  5641 

1572.6 

.233  6796 

2783.5 

.144  0401 

12064 

43 

.027 

1652 

1101.0 

.500  0346 

158J  ; 

.250  4884 

2819.7 

.218  5102 

12773 

44 

.033 

7885 

1106.7 

.509  5746 

'595^8 

.267  5170 

2856.8 

•297  4963 

'357- 

45 

6.040 

4457 

1112.4 

6.519  1850 

1607.7 

7.284  7712 

2894.8 

9.381  5820 

14476 

40 

.047 

1372 

1118.1 

.528  8669 

1619.6 

.302  2571 

2934.1 

•47'  47" 

15510 

47 

.053 

8634 

1123.9 

.538  6216 

1631.8 

.319  9810 

2974.2 

.568  0247 

16704 

48 

.060 

6246 

1 129.8 

.548  4499 

1644.2 

•337  9494 

3015.6 

.672  3106 

18096 

49 

.067 

4212 

"35-7 

•558  353° 

1656.8 

.356  1692 

3058.1 

.785  6758 

'974' 

50 

6.074 

2535 

1141.7 

6.568  3320 

1669.6 

7^374  6475 

3101.7 

9.909  8535 

21715 

51 

.081 

1 21 9 

1147.7 

.578  3881 

1682.A 
1695.6 

•393  39'8 

3146.8 

10.047  1256 

24127 

!  52 

.088 

0269 

1153-8 

.588  5227 

.412  4099 

3'93-o 

.200  5829 

27144 

53 

.094 

9687 

1160.0 

.598  7368 

1708.9 

•43'  7097 

3240.7 
3289.9 

•:74  5584 

31023 

54 

.101 

9479 

1166.3 

.609  0317 

1722.6 

•45'  2999 

•575  3986 

36197 

55 

6.108 

9647 

1172.6 

6.619  4086 

1736.4 

7.471  1892 

3340-3 
3392.6 

10.812  9421 

4345° 

50 

.1:6 

0196 

1179.0 

.629  8689 

'75°^3 

•49'  3870 

11.103  6719 

57 

.123 

1131 

1185.4 

.640  4141 

1764.5 

.511  9029 

3446.5 

11.478  48S0 

58 

.130 

*4S5 

1192.0 

.651  0455 

I779-0 

.532  7472 

3502.1 

12.006  7617 

50 

.137 

4173 

1198.6 

.661  7645 

1793.8 

•553  9305 

3559^6 

12.909  8516 

00 

6.144 

6289 

1205.3 

6.672  5724 

1808.8 

7.575  4640 

3618.7 

. :j 

m 


TABLE  VII. 

For  findlnfj  tlic  True  Anonuily  in  si  I'araliolii'  Orbit  when  r  is  nearly  180°. 


w 

.^0 

IMIT. 

to 

Ao 

Dim 

10 

t^ 

_      —1 
Diff.   1 

o 

/ 

f             H 

H 

0          / 

/       ti 

II 

0 

t 

1         II 

II 

153 

0 
5 
10 
15 
20 
25 

3  2  3-°9 
19.74 
16.43 
13.17 

9-95 
b.T! 

3-35 
3-3' 
3-26 
3.22 
3.18 
3.14 

100     0 

5 
10 
15 
20 
25 

I     6.70 

5-33 
3-97 
2.64 

'•33 

0.04 

1.36 

'-33 
1.31 
1.29 
1.26 

165 

0 

10 
20 
30 
40 
50 

0  15.85 
14.98 

14.16 

13.38 

12.63 
11.91 

0.87 
0.82 
0.78 

0.75   1 

0.72 

0.69 

153 

30 

3     3-63 

160   30 

0  58.78 

166 

0 

0   11.22 

0.65   1 
0.62 

35 
40 

0-54 
»  57-49 

3.09 

3.C5 

35 
40 

57-54 
56.31 

1.24 
1.23 

10 
20 

10.57 
9-95 

45 
50 
55 

54.48 
51.51 
48.58 

3.01 
2.97 
2.93 
2.89 

45 
50 
55 

55.11 

53-93 
52.77 

1.20 
1.18 
1.16 
1. 14 

30 
40 
50 

9.36 
8.80 
8.26 

0.59 
0.56 
0.54 
0.51 

156 

0 

5 

10 

15 

2  45.69 
42.84 
40.03 
37.26 

2.85 
2.81 

2.77 

161      0 

5 
10 
15 

0  51.63 

SO- 50 
49.40 

48.32 

1. 13 

l.IO 

1.08 
1.06 

167 

0 

10 
20 
30 

0     7-75 
7.27 
6.81 
6.37 

0.48 
0.46 
0.44 

20 

34'53 

2.73 

20 

47.26 

40 

5.96 

0.41 

25 

31-83 

2.70 
2.66 

25 

46.21 

1.05 

I.02 

50 

5-57 

0.39 
0.37 

15G 

30 

2  29.17 

2.62 
2.58 
2.54 

2.48 
2.44 

161    30 

0  45.19 

168 

0 

0     5.20 

0.36 

0.33 
0.31 
0.30 
0.28 
0.26  1 

35 
40 
45 
50 
55 

26.55 
23.97 
21.43 
18.92 
16.44 

35 
40 
45 

50 

55 

44.18 
43.19 
42.22 
41.26 
40.33 

1.01 

0-99 
0.97 
0.96 
0.93 
0.92 

10 
20 
30 
40 
50 

4.84 
4.51 
4.20 

3-90 
3.62 

157 

0 
5 

2  14.00 
II.59 

2.41 

162     0 

5 

0  39.41 
38.51 
37.62 

0.90 

169 

0 
10 

0     3.36 
3.11 

0.2s 

10 

9.22 

2.37 

10 

0.89 

20 

2.88 

0.23 

15 
20 
25 

6.89 

4-58 
2.31 

2-33 
2.31 

2.27 
2.23 

15 
20 
25 

36.75 

35-90 
35.06 

0.87 
0.85 
0.84 
0.82 

30 
40 
50 

2.66 
2.46 

2.27 

0.22 
0.20 
0.19 
0.18 

157 

30 
35 

a     0.08 
I   57.89 

2.19 

162   30 

35 

0  34.24 
33-43 

0.81 

170 

0 
10 

0     2.09 
1.92 

1 

0.17  1 
0.16  i 

40 

55-72 

2.17 

40 

.    32.64 

0.79 

0.78 
0.76 
0-75 
0.73 

20 

1.76 

45 

53-57 

2.15 

45 

31.86 

30 

1.62 

0.14  , 

50 
55 

51.46 
49-39 

2. II 

2.07 
2.04 

50 
55 

31.10 
30-35 

40 
50 

1.48 
1-35 

0,14 
0.13 
0.12 

158 

0 
5 

I  47-35 
45.34 

2.01 

163     0 

5 

0  29.62 
28.90 

0.72 

171 

0 
10 

0     1.23 
1.12 

0..1 

10 
15 

43-35 
4«-39 

1.99 
1.96 

10 
15 

28.20 
27.51 

0.70 
0.69 

0.68 
0.67 
0.65 

20 
30 

I.OZ 

0.93 

O.IC    I 

0.09  1 

20 
25 

39-47 

37-57 

1.92 
1.90 
1.87 

20 
25 

26.83 
26.16 

40 
50 

0.84 
0.76 

0.09 

0.08 

0.08  ! 

138 

30 

"   35-70 

1.83 

i.8i 
1.78 
1.76 

163   30 

0  25.51 

0.63 
0.63 
0.61 
0.60 

172 

0 

0    0.68 

35 
40 
45 
50 

I3-87 
32.06 
30.28 
28.52 

35 
40 
45 
50 

24.88 
24.25 
23.64 
23.04 

10 
20 
30 
40 

0.61 
0.55 
0.49 
0.44 

0.07 
0.06 
0.06 

0.05 

55 

26.80 

1.72 

55 

22.45 

0.59 

50 

0.39 

0.05 

1.70 

0.57 

0.04 

139 

0 

I  25.10 

164     0 

0  21.88 

173 

0 

0     0.35 

5 

23.4- 

1.67 
1.65 
1.62 

6 

21.31 

0.57 

10 

0.31 

0.04 

10 

21.78 

10 

20.76 

0.55 

20 

0.27 

0.04 

15. 

20.16 

15 

20.22 

0.54 

30 

0.24 

0.03 

20 

18.57 

1-59 

20 

19.69 
19.18 

0.53 

40 

0.21 

0.03 

25 

17.00 

1-57 

25 

0.51 

50 

0.19 

0.02 

'•55 

0.51 

0.03  1 

159 

30 

1   15.45 

164  30 

0  18.67 

174 

0 

0     0.16 

1 

35 

'3-94 

1.51 

35 

18.17 

0.50 
0.48 

175 

0 

0.07 

0.09  1 

40 

12.44 

1.50 

40 

17.69 
17.21 

176 

0 

0.02 

0.05 

46 

10.97 

'•47 

45 

0.48 

177 

0 

0.01 

O.OI 

50 

9-53 

1.44 

50 

16.75 

0.46 
0.46 

178 

0 

0.00 

O.OI 

55 

8.10 

1.43 

55 

16.29 

179 

0 

0.00 

0.00 

1.40 

0.44 

0.00 

160 

0 

I     6.70 

165     0 

0  15.85 

180 

0 

0     0.00 

1 

1 

611 


TABLE  VIII. 

For  findiri};  thi;  Tinu'  from  the  Pcriliclion  in  a  Parabolic  Or])it. 


log  A' 


o      t 

0  0 
30 

1  0 
30 

2  0 
30 

3  0 

;!0 

4  0 

30 
3     0 

30 

0    0 

30  j 

7  0  I 

30! 

8  (I  ' 

30  i 

9  0 
30 

10  0  { 
30 

11  0 
30 

12  0 

30  I 

13  0 
30 

14  0 

30  ! 

15  0 

30 

16  0 
30 

17  0 

30 

18  0 
30 

19  0 
30 

20  0 
30 

21  0 

30 

22  0 
30 

23  0 

30 

24  0 

30 

25  0 
30 

26  0 

30 

27  0 

30 

28  0 
30 

29  0 

30 


5763 
i749 

5707 
5638 

5541 

54' X 

5266 

5087 
4881 
4647 
4386 

4097 

378" 
3437 
3066 
2668 
2243 
1791 

1311 

0805 
0271 
97  u 
9124 
8510 

7869 
7201 
6507 
5786 

5039 
4266 

3466 
2641 
1789 
091 1 

0008 
9079 

8125 

7>4S 
6140 
S109 
4054 
2973 

1868 

0738 
95S4 
8405 
7202 
5975 

0.022  4724 
.022  3449 
.022  2151 
.022  0829 
.021  9484 
.021  8116 

0.021  6726 

.021  5312 

.021  3876 

.021  2418 

.021  0938 

.020  9436 


0.025 
.025 
.025 
.025 
.025 
.025 

0.025 
.025 
.025 
.025 
.025 
.025 

0,025 
.025 
.025 
.025 
.025 
.025 

0.025 
.025 
.025 
.024 
.024 
.024 

0.024 
.024 
.024 
.024 
.024 
.024 

0.024 
.024 
.024 
.024 
.024 
.023 

0.023 
.023 
.023 
.023 
.023 
.023 

0.023 

.023 
.022 
.022 
.022 
.022 


11 


30  0  I  O.02O  7913 


Diir. 

V 

0      / 

30    0 

14 

30 

4^ 

31     0 

69 

30 

96 

32     0 

124 

30 

152 

33    0 

179 

,30 

206 

34    0 

261 

30 

35    0 

289 

30 

316 

30    0 

344 

30  ! 

5^i 

37     0  i 

398 

30 

425 

38     0 

452 

30 

480 

506 

39    0 

30  1 

534 

40    0  ! 

560 

30 

587 

41     0 

614 

30! 

641 

1 

668 
694 

42  0  i 
30 

43  0 

721 

30 

747 

44     0 

773 

30 

800 

45     0 

825 

30 

852 

46    0 

878 

30  f 

903 

47    0  1 

929 

30 

954 

48    0  { 

9S0 

30 

IC05 

49    0 

1031 

30 

1055 

50    0 

1081 

30 

1 105 

51     0 

1 1  30 

30  1 

1 1  54 

52    0 

;    1179 

.sol 

!  1203 

53     0  1 

1227 

30  1 

1251 

54     0  1 

1275 

30  ! 

1298 

55     0 

1322 

30 

\  5345 

56     0 

1368 

30 

1390 

57     0 

1414 

30 

,  1436 

58    0 

145S 

30 

1480 

39    0 

1502 

30  1 

'5^3 

60     0 

K'g  -V 


0.020 
.020 
.020 
.020 
.020 
.019 

0.019 
.019 
.019 
.019 
.019 
.018 

0.018 
.018 
.018 
.018 
.018 
.017 


7913 
6368 

4802 

3^'; 
1607 

9979 

8330 

6662 

4974 
3267 
1540 
9795 

8030 
6248 
4448 
2629 

0794 
8941 

0.017  707^ 

.017  5186 

.017  3283 

.017  1365 

.016  9432 

.016  7483 

0.016  5520 
.016  3542 
.016  1550 

•°i5  9545 
.015  7526 

•o>5  5495 

0.015  3450 

.015  1394 

.014  9326 

.014  7247 

.014  5157 

.014  3057 

0.014  0947 
.013  8827 
.013  6698 
.C13  4561 
.013  2416 
.013  0263 

0.012  8103 
.012  5936 
.012  3764 
.012  1585 
.011  9402 
.011  7215 

0.011  5024 
.011  2829 
.011  0632 
.010  8432 
.010  6231 
.010  4029 

0.010  1827 
.009  9625 
.009  7424 
.0C9  5225 
.009  3028 
.009  0834 

0.008  8 644 


Diir. 


545 
566 

587 
608 
628 
649 

668 
688 

737 
727 

745 
765 

782 
800 
819 

869 

886 
903 
918 
933 
949 
963 

978 
992 
2005 
2019 
2031 
2045 

2056 
2068 
2079 
2090 
2100 
2110 


2120 
2129 

2137 
2145 

=  153 
2160 

2167 
2172 
2179 
2183 
21S7 
2191 

2195 
2197 
2200 
2201 
2202 
2202 

2202 
2201 
2199 
2197 
2194 
2190 


o       / 

60    0 

30 
01     0 

30 

62  0 

30 

63  0 

30 

04     0 

30 

65  0 

30 

66  0 

30 

67  0 

30 

68  0 

30 

69  0 

30 

70  0 
30 

71  0 

30 

72  0 

30 

73  0 

30 

74  0 
30 

75  0 

30 

76  0 

30 

77  0 
30 

78  0 
30 

79  0 

30 

80  0 

30 

81  0 
30 

82  0 
30 

83  0 
30 

84  0 
30 

83     0 
30 

86  0 
30 

87  0 
30 

88  0 
30 

89  0 
30 

90  0 


loK  y 


864A 
645J 

4*77 
2103 

9934 
7774 

5621 

3477 
'343 
9220 
7108 
5008 

2922 
0849 
8792 
6750 

47^5 
2717 

0.005  07*9 
.004  8760 
.004  681 1 
.004  4884 
.004  2980 
■004  1 1 00 

0.003 
.003 
.003 
.003 
.003 
.003 


o.ooS 
.008 
.008 
.008 
.007 
.007 

0.007 
.007 
.007 
.006 
.006 
.006 

0.006 
.006 
.005 
.005 
.005 
.005 


0.002 
.002 
.002 
.002 
.002 
.002 

O.OOl 

.001 
.001 

.001 
.001 

.001 

0.00 1 
.001 

.000 

.000 
.000 
.000 

0.000 
.000 
.000 

.000 

.000 
.000 

0.000 
.000 
.000 
.000 
.000 
.000 


9245 
7416 
5613 
3839 
2094 

0380 

8698 
7049 

5433 
3854 
2311 
0806 

9341 
79'7 
6535 
5196 

3903 
2656 

1458 
0309 
9211 
8166 

7«75 
6240 

5364 
4546 

379° 
3096 
2468 
1906 

1413 
0990 
0639 
0363 
0163 
0041 


Diff. 


1186 
2181 

2174 
2169 

2lfio 

*I53 

2J44 

a' 34 
2123 
2112 

2100 
2086 

2073 
2057 
2042 
2025 
200X 
1988 

1969 

1949 

1927 
190^ 

188c 
1855 

1829 

1803 
'774 
'745 
1714 
1682 

1649 
1616 
'579 
'543 

1505 
1465 

142.4 
1382 
1339 
1293 

1247 
119S 

"49 

1098 

1045 

991 

935 
876 

818 

756 
694 
628 
562 
493 
423 

35' 

276 

200         ; 

122 

4'  : 


0.000  0000 


1        •' 

1  100 
i           3 
101 

3 

102 

1           3 
1  103 
i          3 
104 

3 

105 

3 
i  106 

107 

108 

109 

3 
110 

3 

111 

112 

3 
113 

3 

114 

31 
115 

116 

3 

117 

3 
118 

3( 
119    ( 

3( 

120 

G12 


8644 
645S 

4*77 

2I05 

9934 

7  .5^^' 

7  3477 

7  1343 

i6  9220 

16  7108 

16  5008 


Dlff. 


2186 
2181 
2174 
2169 
2160 
*>53 

2J44 
2134 
2123 
2112 

2100 
2086 

2073 
2057 
2042 
2025 
200S 
1988 

1949 
1927 
1904 
1 8  8c 
1855 

1829 
1803 
1774 
1745 
1714 
1682 


1649 
I6I6 

1579 
1543 
1505 
1465 

142.4 
1382 

'339 
1293 
1247 
119S 

1149 

1098 

J  045 

99' 

935 

876 

818 

756 
694 
628 
562 
493 
423 

35' 
276 
200 
122 
41 


«0     0 
30 

01  0 
30 

02  0 
30 

03  0 

30 

04  0 
30 

05  0 

30 

00    0 

30 

or   0 

30 
08     0 

30 

00     0 

30 

100  0 
30 

101  0 

30 

102  0 

30 

103  0 
30 

104  0 

30 

105  0 

30 

106  0 
30 

107  0 

30 

108  0 
30 

100     0 
30 

110  0 
30 

111  0 

30 

112  0 
30 

113  0 

30 

114  0 

30 

115  0 
30 

IIG     0 

30 

117  0 

30 

118  0 
30 

110     0 

30 


TABLE  VIII. 

For  tintlint;  the  Time  from  tin-  l'i;rilirli(»ii  in  :i  I'.inibolio  Orbit. 


lug  ^V 


0.000  0000 

9.999  9876 

999  9507 

•999  ****93 

•999  *'039 

•999  6944 


9.999 
•999 
•999 
•999 
.998 
.998 

9.998 

•997 
•997 
•997 
.996 
.996 

9.996 

•995 
•995 
•994 
•994 
•993 

9-993 
.992 
.992 
.991 
.991 
.990 


5613 
4046 
2246 
0215 

7955 
5468 

2757 
9824 
6669 

3297 
9708 
5906 

1891 
7666 

S596 

3755 
8712 

3470 
8031 
2397 
6570 
0553 
4347 


9.989  7956 

.989  1380 

.988  4622 

.987  7685 

.987  0571 

.986  3281 

9.985  5819 

.984  8186 

.984  0385 

.983  2418 

.982  4288 

.981  5996 

9.980  7545 

•979  «93i* 

•979  °'77 

.978  1264 

.977  2202 

.976  2993 

9.975  3640 
•974  4'45 
•973  45'° 
•97a  4739 
•97'  4833 
.970  4796 

9.969  4629 
.968  4337 
.967  3920 
.966  3382 
.965  2726 
.964  1954 


niiT. 


120    0 1   9.963  1069 


124 

369 
614 

854 
1095 
1331 

1567 
1800 
2031 
2260 
2487 
2711 

*933 

3155 

3372 

3589 
3802 

4015 

4225 

4638 
4841 

S043 
5242 

5439 
5634 
5827 
6017 
6206 
6391 

6576 
6758 
6937 
7114 
7290 
7462 

7633 
7801 
7967 
8130 
8292 
8451 

8607 
8761 

89'3 
9062 
9209 
9353 

9495 
9635 

977' 

9906 

10037 

10167 

10292 
10417 
10538 
10656 
10772 
10885 


l*         i 

0      t 

120    0 

;!() 

121     0  ! 

:io  1 

122    0  ! 

30  i 

123    0 

;!0 

124    0 

."0 

125    0 

30 

120    0 

30  1 

127     0 

30 

128     0 

30 

120    0 

30  ! 

130    0  '\ 

30  ! 

131     0  ! 

30 ; 

132     0 

30 

133     Ol 

30  i 

134     0  i 

30 

135     0  { 

30 ; 

130     0  1 

30 

137     0 

30  ' 

138     0  i 

30  , 

130     0 

30 

140     0  1 

30 

141     0 

30    : 

142     0  1 

30  [ 

143     0 

30  1 

144     0 

30  i 

145     0  i 

30  j 

140     0  1 

30 

147     0  \ 

30 

148     0 

30  ' 

140     0 

30  1 

150    0 

1 

log  -V 


9.963    1069 
.962   0074 


.960 

•959 
.958 

•957 


8971 
7764 
6454 
5046 


9.956  3542 
•95  5  «945 
•954  0^58 
.952  8483 
.951  6624 
.950  4684 

9.949  2666 
.948  0573 
.946  840S 
•945  6174 
•944  3875 
•943   1 5' 3 


9.941 

.940 

•939 
.938 
.936 
•935 

9-934 
■933 
•93' 
•93° 
.929 
.927 

9.926 
.925 
.924 
.922 
.921 
.920 

9.918 
.917 
.916 
.915 
.913 
.912 

99" 

.909 
.908 
.907 
.906 
.904 

9.903 
.902 
.901 
.899 
.898 
.897 

9.896 
.894 
.893 
.892 
.891 
.890 


9092 
6615 
4085 
1506 
8881 
6213 

3506 
0763 
7987 
5183 
1353 
9501 

6630 

3745 
0848 

7943 
5°35 
2126 

9220 
6321 
343  3 
0559 

7703 
4870 

2062 
9283 
6538 

383' 
1 164 

854^ 

5969 

3449 

0985 
8582 
6243 
3972 

'774 
9652 
7610 

5652 
3782 
2004 


9.889  0321  j 


613 


Din. 


0995 

1 103 
1207 
I  310 
1408 

1504 

1597 
1687 

'775 
1859 
1940 

2018 

2093 
2165 
2234 
2299 
2362 
2421 

2477 
2530 

1579 
2625 
2668 

2707 

1743 
2776 
2804 
2830 
2852 
2871 

2885 

2897 
2905 
2908 
2909 
2906 

2S99 
2888 
2874 
2856 
2833 
2808 

2779 
2745 
2707 
2667 
2622 
1573 

2520 
2464 
2403 

^3  39 
2271 
2198 

2122 
2042 
1958 
1870 
1778 
1683 


loK  .V 


o       I 

50  0  ' 

30 

51  0 
30 

52  0 

liO 

53  0 

30 

54  0 

30 

55  0  ' 

30  I 

50     0 

30  i 

57  0  ' 
30  I 

58  0 
30 

50     0 

30 
00     0 

30 
61     0 

30 

02  0 

30 

03  0 
30 

04  0 

30 

05  0 

30 
00     0 

30 
67     0 

30 

08     0 

30 
00    0  I 
30  ! 

70  0 

30 

71  0 

30 

72  0 
30 

73  0 

30 

74  0 

30 

75  0 
30 

70     0 


77 

30; 
0  ' 

30 

78 

0 

30  ; 

70 

0 ; 

30 

80 

1) : 

9.889 

.887 

.886 
.885 
.884 
.883 

9.882 
.881 

.880 
•879 
•877 
.876 

9.875 

■874 
.873 
.872 
.87, 
.871 

9.870 
.869 
.868 
.867 
.866 
.865 

9.864 
.864 
.863 
.S62 
.861 
.86i 

9.860 
.859 
.858 
.858 
.857 
.857 

9.856 
.855 

•855 
.854 
,854 

•853 

9.853 

•853 
.852 
.852 
.851 
.851 

9.851 

.850 
.850 
.850 
.850 
.850 


0321 

8738 
7ii9 
5887 
4627 
3481 

2455 
'551 
0775 
0129 
9616 
9242 

9010 
8922 
8984 
9198 
9569 
0099 

0792 
1652 
2683 
3.S86 
5266 
6827 

8570 
0500 
2620 
4932 

7439 
0145 

3053 
6164 
9482 
3010 

6750 
0704 

4875 
9266 
3878 
8714 

3775 
9065 

4584 
0335 

6319 

2538 
8994 
5687 

2620 

9794 
7209 
4868 
2770 
0917 


9.849  9309 

•849  7948 
.849  6833 
.849   5966 

•849  5346 
.849  4974 

9.849  4850 


Diir. 


11583 

"479 
11372 
1 1  260 
1  I  1 46 
I  1026 

10903 
10777 
10646 
I  05  I  3 
10374 
10232 

10088 

9938 
9786 
9629 

9470 
9307 

,  9140 
8969 

8797 
8620 
8439 

'  8257 

I 

8070 
7880 
7688 

7493 
7294 
7092 

6889 
6682 
6472 
6260 
6046 
5829 

5609 
5388 
5164 

4939 
4710 
4481 

4249 
4016  j 

3781 
3  544  i 
33°7  ■ 
3067 

2826  i 
2585  '■ 

234'  : 
2098  I 

'853  ' 
1608 

1361 

'"5 

S67 
620 
372 
124 


TABLE  IX. 

For  fiiuling  iho  True  Aiimiiiilv  or  tlie  Tiiiiu  liuiu  iliu  I'oriliclion  in  Orlatx  of  grcnt  eccentricity. 


X 

.1 

o 

// 

() 

0.00 

1 

0.00 

'Z 

0.0 1 

.1 

O.OJ 

4 

0.12 

5 

0.23 

0 

0.39 

7 

0.62 

H 

0.93 

« 

'•33 

10 

1.82 

11 

2.42 

la 

3.14 

Vl 

3  99 

14 

4.99 

15 

6.13 

k; 

7-43 

17 

8.90 

IH 

10.55 

10 

12.40 

'M 

14-45 

ai 

16.70 

'M 

19.18 

•Z3 

21.89 

'Zl 

24.83 

25 

28.03 

2ft 

31.48 

27 

35.20 

28 

39.19 

29 

4347 

»0 

48.04 

:u 

52.91 

i.1 

58.09 

•M 

63.59 

34 

69.42 

35 

75'57 

36 

82.07 

37 

88.92 

38 

96.12 

39 

103.68 

40 

in.6i 

41 

119.92 

42 

128.62 

43 

137.70 

44 

147.18 

i:» 

'57-05 

40 

i;'7.34 

47 

178.14 

48 

189.16 

49 

200.71 

50 

212.69 

51 

225.10 

52 

i37->5 

53 

251.  15 

54 

265.01 

55 

279.-.  I 

5ft 

293.88 

57 

309.02 

58 

324.6s 

59 

340.70 

00 

357.26 

iiiir. 


0.00 
0.01 
0.04 
0.07 
o.  n 

0.16 

0,23 
0.31 
0.40 
0.49 

0.60 
0.77. 
0.85 
1.00 
1. 14 

1.30 

1-47 
1.65 
1.85 

2.05 

2.25 
2.48 
2.71 
2.94 

3.20 

3-45 
3-7* 
3-99 
4.28 

4-57 

4.87 
5.18 

5-50 
5.83 
6.15 

6.50 
6.85 
7.20 
7.56 
7-93 
8.3, 
8.70 
9.08 
9.48 
9.87 

10.29 
10.70 
11.12 
11.55 
11.98 

12.41 
12.85 

13-30 
13.76 
14.20 

14.67 
15.14 
t5.6o 
ib.c'^ 
16.56 


Hill. 


0.000 
0.000 
0.000 
0.000 
0.000 

0.000 
0.000 
0.000 

0.000 
0.000 

0.000 
0.000 

0.000 

0.000 

0.00 1 

0.001 

0.002 

0.002 

0.003 
0.004 

0.005 
0.006 
0.008 
0.010 

0.01  2 

0.014 
0.017 
0.010 
0.025 
0.030 

0.035 

0.041 

0.047 
0.055 

0.064 

0.073 

0.084 
0.096 
0.109 
0.123 

0.139 

0.156 

0.175 

0.196 

0.218 
0.243 

0.269 
0.298 

0.328 

0.361 

0.397 
0.436 

0-477 
0.521 

0.567 

0.617 

0.671 

0.727 
0.787 
0.851 

0.919 


.001 

.000 

.001 
.COl 

.001 

.001 

.00s 
.002 

.002 
.002 

.003 
.003 
.005 
.005 

.005 

.006 
.006 

.008 

.009 
.009 

.011 

.012 

.013 

.014 
.016 

.017 
.019 

.021 
.022 

.025 

.026 
.029 

.030 

■°Ti 
.036 

.039 

.041 

.044 

.046 
.050 

■054 

.056 
.060 
.064 

.068 


bU 


0.000 
0.000 

0.000 

0.000 
0.000 

0.000 

0.000 
0.000 
0,000 

0.000 

0.000 

0.000 

0.000 
0.000 
0,000 

0.000 
0.000 
0.000 
0.000 
0,000 

0,000 

0,000 
0,000 
0,000 
0.000 

0.000 
0.000 

0.000 
0,000 

0.000 

0,000 
0.000 

0.000 

0.000 
0.000 

0.000 
0.000 
0.000 
o.ooo 
0.000 

0.000 
0.000 
0.000 

0.000 

0.000 
0.000 

0.000 

0.000 
0.000 
0.000 

0.000 
0.000 

O.OOI 

0.001 

O.OOI 
O.OOI 

0.002 
o.ooz 

0,002 

0.002 
0.003 


11' 

Dill. 

c 

1 

It 

:      0.000 

0.000 

'     0.000 

0.000 

0.000 

o.oco 

0.000 

0,000 

0,000 

0.000 

0.000 

0.000 

0,000 

0.000 

c.ooo 

0.000 

0,000 

0.000 

0.000 

0.000 

0.000 

0.000 

0.000 

0.000 

0.000 

0.000 

0.000 

0.000 

O.OOI 

0.000 

O.COI 

0.000 

O.OOI 

.000 

0.000 

0.002 

.001 

0.000 

0.002 

.000 

0.000 

0.003 

.001 

.001 

0.000 

0.004 
0.005 

.001 

0.000 
0.000 

0.006 

.001 

0.000 

0.008 

.002 

0.000 

0.010 

.002 

.002 

0.000 

0.012 

0.000 

0.014 

.002 

0.000 

0.017 

.003 

0.000 

0.020 

.003 

0.000 

0.024 

.004 

0.000 

004 

0.028 

0.000 

0.033 

.005 
.006 
.006 

0.000 

0.039 
0.045 

o.o-^o 
0.000 

0.052 

.007 

.008 

0.000 

0.060 

.008 

0.000 

0.068 

0.000 

0.078 

.010 

0.000 

0.088 

.010 

0.000 

O.I  00 

.012 

.013 

0.000 

O.I  13 
0.127 

0.000 

.014 

0.000 

0.142 

.015 

0.000 

0.159 

0.177 

.017 
.018 

.020 

o.coo 

0.000 

0.197 

o.coo 

0.219 

.022 

0.000 

0.242 
0.267 

0.294 

.023 

.025 
.027 

.029 

0.000 
0.000 
0.000 

0.323 

0.354 
0.388 
0.424 

0.462 

0.000 

.031 
.034 

.036 
.038 

0.000 

o.coo 
o.coo 
o.coo 

.040 

0.502 

0.000 

0.546 

.044 

.046 

0.001 

0.592 

O.OOI 

0.641 

.049 

O.COI 

0.693 

.052 

.056 

O.OOI 

0.749 

0.002 

cat  ccccntiii'ity. 


o.ooo 
o.ooo 
o.oco 
o.ooo 
o.ooo 

o.ooo 

0.000 

o.ooo 
o.ooo 
o.ooo 

o.ooo 
o.ooo 

0.000 

o.ooo 
o.ooo 


0.000 

)00 

0.000 

>0I 

0.000 

300 

0.000 

)0I 

0.000 

501 

0.000 

301 

0.000 

301 

0.000 

302 

0.000 

302 

0.000 

302 

0.000 

DO  2 

0.000 

30  3 

0.000 

JO  3 

0.000 

004 

0.000 

304 

0.000 

30  5 

0.000 

300 

O.O-'O 

306 

0.000 

'°l 

0.000 

D08 

0.000 

308 

0.000 

3IO 

0.000 

DIO 

0.000 

312 

0.000 

31-? 

0.000 

314 

0.000 

3«5 

0.000 

"7 

o.ooo 

3l8 

0.000 

320 

0.000 

322 

0.000 

323 

o.oco 

D25 

0.000 

327 

0.000 

329 

0.000 

331 

0.000 

=34 

o.ooo 

'^^ 

0.000 

35S 

0.000 

340 

0.000 

H4 

0.00 1 

34b 

0.00 1 

349 

O.COI 

552 

O.COI 

3Sb 

0.002 

TABLE  IX. 

Kor  liiidinK  tlie  Triio  .Vnonmly  or  tlie  Tiiiu!  t'roni  the  I'tiilielioii  in  Orhitnof  urcnt  epcpntricify. 


»l 

0'^ 

m 
111 

(»r 

(M 

ou 

70 

ri 
7a 
73 
74 

75 

70 

77 
78 
71) 

80 
81 

S'Z 
83 
81 


8U 

iX) 
91 
1W 
»3 
01 

05 
UU 
97 
98 
99 

100  U 

101  (I 

;!() 

102  (I 

30 

103  0 

30 
101    0 

30 
103    0 

30 

lOG    0 

30 

i  107     0 

30 
i  108    0 

30 

109    0 


Dlff. 


357.26 

409.x  6 
42X.38 

44;40 
4(16.92 
4S6.96 

507.51 
52S.58 

550-«7 
572.29 

5^4  94 
61S.12 
64. .85 

666.13 
690.96 
716.34 
742.29 
768.81 

795.90 

823.57 
851.84 
880.70 
910.16 

940. -.3 

970.92 

1002.24 

5034,20 

1066.81 

1 100.08 
1 134.02 
1168.64 
1203.95 
1239.97 

1276.72 
1314.21 

'3  52-45 
1391.46 
1431.27 

1471.88 
1492.50 
>5'333 
•534-3» 
«555'64 
1577.12 

1598.82 
1620,75 
1642.91 
1665.30 
1687.93 
1710.80 

1733.92 
1757.28 
1780.90 
1804.77 
1828.90 
1853.30 

1877.97 


17.04 

>7-54 
18.02 
18.52 
19.02 

19.52 

20.04 
20.55 
21.07 
21.59 

22.12 
22.65 
23.18 

a3'73 
24.28 

24.83 

*5-3X 
aS-95 
26.52 
27.09 

27.67 
28.27 
28.86 
29.46 
30.07 

30.69 

3'-32 
31.96 
32.61 
33-i7 

33-94 
34.62 

35-3' 
36.02 

36-75 

37-49 
38.24 
39.01 
39.81 
40.61 

20.62 
20.83 
21.05 
21.26 
21. 4S 
21.70 

21.93 
22.16 
22.39 
22.63 
22.87 
23.12 

23.36 
23.62 
23.87 
24.13 
24.40 
24.67 


0.919 

0,990 
1.066 
1. 145 
1.230 

1. 318 
1.411 
1. 510 
1.61  3 

1.721 
1.8,5 

'■954 
2.078 
2.209 
2-345 

2.4S8 
2.637 

2-793 
2.95  b 
3.125 

3.302 
3.4X6 
3.678 
3.878 
4.087 

4.303 
4.529 
4.764 

5.008 
5.262 

5  527 
5.801 
6.087 
6.385 
6.694 

7.016 

7-350 
7.698 

8.060 

8-437 

8.829 
9.032 
9.238 

9-449 
9.664 
9.883 

10.108 

'°-337 
10.570 
10.809 
11.053 
11. 302 

"-557 
11.X17 
12,083 
12.354 
12.632 
12.916 

1 3.207 


IMH. 


071 
076 
079 
0X5 
08X 

093 
099 
103 
108 

"4 

119 
'24 
'3' 

136 

«43 

'49 

'56 
163 
169 
177 
1X4 
192 

2  00 
209 
216 

226 
235 
244 
254 
265 

274 
286 
298 
309 


334 

34i* 
362 

377 
392 


203 
206 
211 


219 

225 

229 
233 
239 

244 
249 

255 

260 
266 
271 
278 
284 
291 


0.003 
0.003 
0.003 

0.004 
0.004 

0.004 
0.005 
0.005 
0.006 
0.006 

0.007 
0.007 
0.008 
0.009 
0.009 

o.oio 
0.01 1 
0.012 

0.0!  3 
O.OI4 

0.0  1  5 
0.016 
0.017 
0.018 
0.020 

0.021 
0.023 
0.024 
0.026 
0.02S 

0.030 
0.032 
0.034 
0.036 
0.038 

0.041 
0.044 
0.047 
0.050 
0.053 


0.058 
0.060 
0.062 
0.064 
0.066 

0.068 
0,070 
0.072 
0.074 
0.077 
0.079 

0.082 
0.084 
0.087 
0.090 
0.093 
o  096 

0.099 


//' 


nin. 


0.749 

0.807 
0.869 
0.935 
1.004 

1.077 
I.I  54 
'■235 
1-321 
1. 41 1 

1.505 
1.605 
1.709 
1.819 
•-934 

2.05s 
2.1X1 
2.314 

2-453 
2.599 

2.752 
2.912 
3.079 
3-255 
3-439 

3.631 

3-«33 
4.044 
4.266 
4.498 

4-74' 
4.996 
5.263 

5.83S 

6.147 
6.471 
6.812 
7.171 
7-549 

7.946 
X.I  52 
8.364 
8.582 
8.805 
9.035 

9.271 

9-5 '3 
9.761 

10.017 
10.280 
10.550 

10.828 
II. 1 14 

1 1.408 
11.711 
12.022 
12.343 

12.673 


!  -058 
.061 

I  .066 
.069 

I   •°73 

•°77 
.0X1 
.0X6 
.090 
.094 

.100 
.104 
.110 
.115 
;  .121 

.126 
i  .133 

•'39 

.146 

:  .153 

i  .160 
.16- 
.1- 

.1.S4 
I   .192 

'     .102 

'     .211 

.222 

.232 

I     -243 

1  -255 
I  -267 
I  .281 
i  .294 
■309 

-324 

•34' 

i   -359 

.378 

i   -397 

I   .206 

I   .212 

.218 

.223 

!  .230 
.236 

I  .242 
.248 
.256 
.263 

.270 
.278 

.286 
.294 

•3°3 
.311 
.321 

.330 


f," 


O.002 
0.002 
0.002 
0.002 
0.002 

0.003 

o  003 
0.003 
0.004 
0.004 

0.004 
0.005 
0.005 
0.00b 
0.006 

0.007 
0.007 
0.008 
0.008 
0.009 

O.OIO 
O.OII 
O.OIl       I 

0.013 
0.014 

0.015 
0.016 
0.018 
0.019 
0.021 

0.023 
0.025 
0.027 
0.029 
0.032 

0.035 

0.038 
0.041 

0.045 
0.049 

0.053 
0.055 

0.05X 
0.060 
0.063 
0.066 

0.069 
0.072 

0.075 
0.078 
0.0X2 
0.0X5 

0.089 
0.093 
0.095 
O.I  02 
0.107 

0.1  12 

0.117 


615 


my. 


TABLE   IX. 

For  finding  tha  True  Anojnaly  or  .lie  Time  from  the  Perilielion  in  Orbits  of  great  eccentricity. 


X 

o 

, 

109 

0 

.•'-It 

110 

0 

.3(1 

111 

II 

;ju 

US 

() 

A 


Diir. 


H'~ 


113    II 

.'ill 

114    II 

:iO 

113    0 

:!0 

lie    0 

;io 

117    0 

:jo 

118     0 

:!0 

110     0 

;iO 

120    1) 

.iO 

121     (I 

:;o 

122    n 

:!() 

123    II 

:u) 

124    fl 

;m) 

125     0 

nn 

120    0 

30 

127     l> 

30 

128     0 

;^o 

129    0 

30 

130    0 

20 

40 

131     0 

20 

40 

132     0 

20 

'10 

133     ' 

20 

40 

134    0 

20 

40 

135    0 

20 

40 

130    0 

1877.97 

1902.91 
I92X.I3 

1953.64 
1979.4+ 

2005.54 

2031.94 
^058. 64 

2085.66 

21  13.00 
2140.66 
2168.66 

2197.00 
2225.69 
2254.73 
2284.13 
2313.91 
2344.06 

2374.60 
2405.54 
2436.88 
2468.64 
2500.83 
253345 

2566.51 

2600.03 
2634.02 
2668.49 
2703.46 
2738.93 

2774.91 
2811.43 
2848.50 
2886.13 
2924.33 
2963.12 

3003.53 
3042.56 
3083.23 
3124.57 
3166.59 
3209.31 

3252.76 
3282.13 

33H-85 
334190 
3372.31 
3403.09 

3434-23 
3465-74 
3497.03 

35--;.9i 
3562.60 

3595-69 

3629.20 
3663.13 
3697.50 
3732.31 
3767.58 
3803,31 

3^39-52 


24-94 
25.22 
25.51 
25.8c 
26.10 
26.40 

26.70 

27.02 

27-34 
27.(11 
28.00 
28.34 

28.69 

29.04 
29.40 
29.78 
30.15 
30-54 
30.94 

3'-U 
31.76 

3'  '9 

3i..02 

33.06 

33-52 

33-99 
34-47 
34-97 
35-47 
35.98 

36.52 

37.07 

37-63 
38.20 
38.79 
39-4« 
40.03 
40.67 
41.34 
42.02 
42.72 
43-45 

29-37 
29.72 
30.05 
30.41 

j<->4 

31-5' 
31.89 
32.28 
32.69 
35.09 
33-51 

33-93 

34-37 
34.81 

35-27 
35-73 
36.21 


3.207 
3.504 
3.808 
4.119 
4.438 
4.764 

5.097 

5-439 

5.789 
6.148 
6.515 
6.892 

7.278 

7.674 

8.080 

8.496 

8.924 

9-363 

19.813 

20.276 

20.751 

21.240 

21.742 

22.258 

22.789 
23.336 
23.X98 

24-477 
25,073 
25.687 

26.320 
26.973 
27.646 
28.341 
29.057 
29-797 
30.562 

31-351 
32.167 
33.011 

33-885 
34.789 

35-725 
36-367 
37.025 
37.699 
38.389 

39-097 

39,822 
40.564 
41.326 
42.108 
42.910 
43-733 

44-576 
45-442 
46.33> 

47-245 


riilT. 


.297 
.304 
.311 
.319 

.326 
-333 
.342 
•350 
-359 
-367 
•377 
.386 

.396 
.406 
.416 
.428 

•439 
.450 

.463 

-47  5 
.489 
.502 
.516 
-53' 

•547 
.562 

-579 
-596 
.614 

•633 

-653 

-673 
.695 
.716 
.740 
-765 

-:'S9 
.816 
,844 
.874 
.904 
.930 

.642 
.658 

-674 
.690 

.708 
.725 

.742 
.762 
.782 
.802 
.823 
-843 

.866 
.889 
.914 


DilT. 


48.183    1    •'-*? 

49.«47   '  '"^ 
50-' 38 


964 
.991 


0.099 

0.102 
0.106 
0.109 
0.1 13 
0.116 

0.110 


o.  24 
0.-28 
O.!  32 

o-'.)7 
0.142 


0.147 

0.152 

0.157 
162 
168 

•74 
180 
1S6 
0.J93 
0.200 
0.207 
0.214 


0.222 
0.230 
0.239 
0.248 
0.258 
0.268 

0.278 
0.289 
0.300 
0.312 

0-325 
0.338 

0.352 
0.367 
0.382 
0.398 
0.415 

0-433 

0.452 
0.465 
0.479 

0493 

0.508 
0.523 


0-539 
°-555 
0.572 
0.590 
c.6og 
0.629 

0.649 
0.669 
0.691 
0.714 
0.738 
0763 

0.788 


.003 
.004 
.003 
.004 
.003 
.004 

.004 
.004 
.004 
.005 
.005 
.005 

.005 

.005 
.005 
.00(1 
.006 
.006 

.006 
.007 
.007 
.007 
.007 
.008 

.008 
.009 
.009 
.010 
.010 
.010 

.011 
.011 
.012 

.013 
.013 
.014 

.015 
.015 
.oi6 
.017 
.018 
.019 

.013 
.014 
.014 

!  -015 
.015 
.016 

.01 6 

.017 


3T8 
.Olr 


i  -024 


Diff. 


12.673 
13.013 

«3-363 
13.724 
14.095 
14.478 

14.874 
15.282 
15.702 
16.135 
16.583 
17.045 

17.522 
18.015 
18.524 

19.050 

•9-594 
20.156 

20.738 
21.339 
21.962 
22.606 
23.273 
23.964 

24.680 
25.422 
26.191 
26.988 
27.815 
28.673 

29.564 
30.489 

3 '-450 

32,448 

33-4*<5 
34-563 

35.685 
36.852 
38.067 

39-331 
40.649 
42.022 

43.452 
44-439 
45-455 
46.500 

47-5:'5 
48.68?. 

49,820 
50.992 
52.199 

53-442 

54-723 
56.042 

57.401 
58.302. 
60.247 
61.736 

63-273 
64.857 

66.491 


340 

35^' 
361 

371 
3*«3 
396 

408 
420 

433 
448 

462 

477 

493 
509 

526 

544 
562 
582 

601 
623 
644 
667 
691 
716 

742 
769 
',/7 
827 
858 
891 

925 
961 
998 

03; 
078 
122 

167 

■■'»5 
264 

318 

373 
430 

016 

045 
075 
107 
138 

172 
207 

243 
281 

319 

359 

401 

445 
489 

537 
584 
634 


C  bill. 


0.II7 
0.122 
0.128 

-  '34 
0.141 
0.148 

0.155 
0.162 

0.170 
0.178 
0.187 
o  196 

0.206 
0.216 
0.227 

0-239 
0.251 

0.264 

0.277 
0.291 
0.306 
0.322 
0-339 
0-357 

0.376 
0.396 
0.417 
0.439 
0.463 
0.4  88 

0.515 
0.544 

0-574 
0.606 
0.640 
0.676 

0.715 
0.757 
0.800 
o.!(46 
o  896 
0-9-19 

1.005 
1.045 
1.087 
1. 130 
1.175 
1.221 


1.8 


1.273 

•-325 
1.379 

i-n6 
1.495 

'-55« 
1.623 
1.692 
1.764 


39 


1.917 

2.000 

2.087 


.005 
.006 
.006 

.007 
.007 
.007 

.007 
.008 
.008 
.C09 
.009 
.010 

.010 
.011 
.012 
.012 
.013 
.013 

.014 
.015 
.016 
.017 
.018 
.019 

.020 
.021 
.022 
.024 

.025 
.027 

.029 
.030 
.032 
034 
.036 
•039 

.042 
.043 
.046 
.o;o 
-053 


i  -040 

I  'J42 

I  -043 

i  -c+i 

I  -04-^ 


.054 
.057 
.059 
.063 
.065 

.061} 
.072 
.075 
.078 

-oi'3 


:,87 


For 

13 
13 

13f 
13! 

14C 
141 

142 
143 
144 
145 


'  4 


140 


3 

4 

5 

147 

1 

1 

1 

21 

,SI 

41 

' 

51 

118 

1 

11 

2( 

.•Ji 

4! 

50 

149 

0 

»1« 


;  eccentricity. 


TABLE  IX. 

For  finding  the  True  Anomaly  or  tlie  Time  from  the  Perihelion  in  Orbits  of  great  eccentricity 


136  n 

2(1 
4(1 

137  0 

20 
■ill 

138  U 

20 

40 

130    0 

20 

40 

140    0 

20 

40 

I  141    0 

J  20 

I  40 

142  0 

j  20 

;;o 

40 

50 

143  0 

10 
20 
oO 
411 
,')0 

144  0 

10 
20 
,'.0 
4'i 
aO 

145  0 

10 

20 

.•!0 

'   40 

0(1 

146  0 

10 

20 

30 

4(1 

■      50 

147  0 

V) 
20 
30 
40 
SO 

148  0 

10 
20 
.'iO 
40 
50 

149  01 


DifT. 


I 


387(1.21 
3913.41 
3951.12 

39*'9-3; 
4.028. n 

4067.42 
4107.28 
4147.72 
4188.75 
4230.38 
4272.63 

4315.52 
4359.06 
4^05.26 
4448.15 

4493-73 
4540.03 

4C87.07 
4610.88 
4634.88 
4659.07 

4683-;' 
4708.05 

4732.84 

4757-84 
4783.05 
4808.46 
4834.10 
4859.95 

4S86.02 
4912.31 
4938.83 
4^05.58 


499^  S 


5019.78 

i;o4;-23 

107+-93 
5102.88 

5131.08 

5159-53 
5188.24 

5217. 2T 

5i4''-45 
5^75-95 
5305-73 
533)-79 
5366.13 

5396.76 
C427.67 
5458.88 
j49°-39 

5^22.20 

5554-33 
5586.77 
5619.52 
5652.60 
5686.01 
5719-75 
5753  ^i 

5788.26 


6  ! 


36.69 

37.20 

37-71 
38.23 
38.76 
39-31 
39.86 
40.44 
41.03 
,1.63 
42.25 
42.89 

43-54 
44.20 
44.89 
45.58 
46.30 
47.04 

23.81 
24.00 

2-  .9 
24.39 

14-59 
24.79 

25.00 
25.21 
25.41 
25.64 

25.X5 
26.07 

26.29 
26.52 
26.75 
26.98 

27.22 
27.45 

27.70 
27.95 
28.20 
28.45 
28.71 
28.97 

29.24 
29.50 

29.78 
30.06 

3°-34 
3"-f'3 
30.91 
31.21 
31.51 
31.81 

3?--<3 
32.44 

3^-75 
33.08 

33-41 
33-74 
34.08 

34-43 


B 


50.138 
51.156 
52.203 
53.280 
54.388 
55.528 

56.702 
57.910 

59-154 
60.436 

6'-757 
63.119 

64.523 
65.971 
67.465 
69.007 
70.599 
72.243 

73-941 

74.811 

75-695 

76.595 

77-509 
78.439 

79-385 
80.347 
81.325 
82.321 

83-333 
84.363 

85.411 

86.478 
87.564 
88.668 

89-793 
90.938 

92.103 
93.290 
94.498 

95-729 
96.982 
98.259 

99-559 
100.884 
102.234 
103.610 
105.012 
IC6.441 

107.897 
109.382 
1 10.896 
112.439 
1 14.013 
1 15.619 

117.256 
1 18.926 
120.631 

122.370 
x  24. 1 44 
I  ■5-955 
127.804 


I 


Din. 


1.018 

1.047 
1.077 
1.108 

1.140 

1.174 

1.20S 

1.244 
1.282 

1.321 
1.362 

1.404 

1.448 
1.494 
1.542 
1.592 

1.0+4 

1.698 

0.87c 

0.884 

0.900 

0.914 

0.9  3D 

0.9+6 

0.962 

0.978 

0.996 

1.012 

1.030 

I.C48 

1.067 

1.086 

1 . 1  0.1^ 

1.125 

1. 145 

1. 165 

187 

20S 

231 
^5  3 
277 
300 


1.325 

,  '-350 
1.376 
I  1.402 
I  J -4-9 
I  I  •45'' 

;  -485 

1.514 

'•543 
,  1-574 
I  1.606 

I  «-637 

i  1.670 

1.705 

'  1-739 

I  '-774 
1. 811 

1.849 


Diff. 


0.788 
0.815  I 

0.843 ! 
0.873 ' 
0.904 1 
0-936 1 
0.969 1 

1.004  ! 
1.041  I 
1.079 
1119  I 
1.161 

1.205 

1.251  : 
1.299  j 

1.350  j 
1.404  : 
1.460 

1.518 
1.549 
1.580 
1.612 
1.645 
'.679  i 

1.714 

J-749  I 
1.786  ' 
1.823 
1.862 
1.901  j 

1.942  I 

1.984  i 
2.026  i 
2.070  j 
?..  116' 
2.162  I 

;..2io  ! 
2.259  I 
2.309 

2.361 

2.414  : 
2..^69  j 

2.516  j 
2.584 
2.643  i 
2.704  , 
2.767 
2.833 

2.900 
2.969 

3.040 

3-"3 
3.188 
3.266 

3  •34'^ 
3.428 

3-513 
3.601 
3.691 
3--'84 
3.881 


oir 


i)iff. 


.027 
.028 
.030 
.031 
.032 
.033 

-035 
.037 
.038 
.040 
.042 
.044 

.046 
.048 
.051 
.054 
.056 
.058 

.031 
.031 
.032 
.033 
■034 
-035 

-Oj5 

•037 

.037 

•039 
.039 
.041 

.042 
.042 
.044 
.046 
.046 


.049 
.05c 
.052 

•053 
.055 

•057 

.058 
.059 
.061 
.063 
.066 
.067 

.069 

.071 
.073 
.075 
.078 
.080 

.082 
.085 
.o?8 
.090 

-093 
.097 


66.491 
68.178 
69.920 
71.718 
73-575 
75-493 

77-475 
79-523 
81.641 

83.830 
86.094 
88.436 

00.860 
93.369 

95-9''7 
98.657 

01-443 
04.331 

07.324 
08.861 
10.427 

12. 022 
13.646 
15.301 

16.986 

i?704 
20.452 
22.233 
24.049 
25.899 

27-/85 
29.707 
31.666 
33.663 
35.698 
37-774 
39.890 
42.048 
44.249 
46.494 
48.784 
51.120 

53-533 
5  5-934 
58.415 
60.947 

<'3-53« 
66.168 

68.860 
71.608 

74-414 
77.280 

80.206 
83.194 

86.246 
89.364 
92  549 
95  804 
99-130 
02.528 

206.002 


1.687 

1.742 
1.798 
1.857 
1.918 
..9S2 

2.048 

2.1-8 

2.189 

2. 264 
2.342 

;  2-424 

2.509 
2.598 
2.690 
'  2.786 
'  2.888 
2-993 

1-537 

i.^ee 

1-595 
1.62^ 
1.655 
1.685 

1.718 

1.748 

:.78i 
1.816 
1.850 
1.886 

1.922 
1.959 
1.997 
2.035 
2.076 
2.116 

2.158 
2.201 
2.245 
2.290 
■  2.336 
2.383 

'  2.431 
2.481 

2.532 

^  2.584 
2.637 
2.692 

2.748 
2.806 
2.866 
2.926 
2.988 
i  3-052 

'3.118 
3.185 

:  3-255 
;  3.326 

3-398 

i  3-474 


Diir. 


2.087 
2.178 
2.274  , 

2-375   I 
2.480  I 

2-59'    I 

2.708 
2.831 
2.960 
3.096 
3.239 
3-390 

3-549 

3-7'7 
3.893 
4.080 

4-277 
4.484 

4.704 
4.819 
4.936 

5-057 
5.181 

5-309 

5-440 
5-575 
5-715 
5-858 
6.005 
6.157 

6.313 
6.473 
6.639 
6.809 
6.984 
7.165 

7-35> 
7-543 
7-740 
7-943 

8-'53 
8.369 

8.592 

8.822 

9.060 

9-3    ' 

9-Si'3 
o.;.i5 

1  .083 
10.359 
10.645 
10.940 
11.244 
11.558 

11.883 
12.218 
12.564 
12.921 
13.291 
'3-673 
14.067 


.091 
.096 
.101 
.105 
.1 1 1 
.117 

.123 
.129 

-136 

•'43 
.151 
.159 

.168 
.176 
.187 
.197 

.207 

.220 
.115 

.117  ! 

.121 

.124 

.128 

.131 

-135 

.140 
.143 
.147 
.152 
.156 

.160 
.166 

.170  I 
-'75  I 

.181  ; 
.186 1 

.192 : 

-'97  I 
.203  I 
.210  , 
.216  ' 
.223  [ 

.230 

.238 

.244 

.251 

.260  J 

.268 

.276 
.286 

-295 
.304 
.314 
-325  ' 

-33  5 
-346 

-357 
.370 
.382 
-394  i 


TABLE  X. 

1 

For 

finding  the  True  A»onialy 

or  the  Time  i'roni  tlie 

Perihelion  in  Elliptic  and  Hyperbolic  Orhits. 

A 

Gllipge. 

ll.viiorlM 

la. 

1 

1 

log  li 

Dill. 

log  a 

logl.Diff. 

los; 

ball  11. 1)1  ir. 

log  Jl 

DIIT. 

log  C 

'"Sl-'^'"^      Lalfn^WIT 

0.000 

0.000 

! 

0.00 

0000 

4 

37 

53 
68 

0.000  0000 

4.23990 

1.778 

0000 

7 
23 
37 
5' 
66 

0.000   0000 

4.23982,,          1 

77«      1 

.01 

0007 

.001  7432 

.24286 

.783 

0007 

9.998    2688 

.23686 

767 

.02 

0030 
0067 

.003  49X5 

.24583 

.788 

0030 

.9<)6   5493 

.23392 

762 

.03 

.005  2659 

.24885 

•794 

0067 

•994  8414 

.23098 

758      , 

.04 

0120 

.007  0457 

.25190 

•799 

0118 

•993   »45o 

.22807 

753     j 

0.05 

0188 

84 

99 
114 

130 

«47 

0.008  8381 

4.25497 

1.805 

0184 

81 

9.991   4599 

4.22  5 1 8„         1 

748     i 

.06 

0272 

.010  C432 

.25806 

.81  • 

0265 

94 
109 
123 

137 

.989  7859 

.22230 

743     ! 

.07 

0371 

.012  4613 

.idl  16 

.816 

0359 

.988   1231 

.21943 

739 

.08 

0485 

.014  2924 

.26427 

.821 

0468 

.986  4711 

.21659 

734     ' 

.09 

0615 

.016  1367 

.26741 

.827 

0591 

.984  8298 

.21376 

730 

O.IO 

0762 

162  °-°i7  9945 

4.27057 

1.833 

0728 

152 
165 
178 
193 
206 

9.983    1992 

4.2io94„         1 

725 

.11 

0924 

T78 

.019  8659 

.27376 

.839 

0880 

.981    5791 

.20815 

720     1 

.12 

1102 

>94 
21 1 

.021     7511 

.27697 

.845 

1045 

•979  9694 

•20537 

716     1 

•«3 

1296 

.023    6503 

.28020 

.85, 

1223 

.978  3699 

.20260 

711     i 

.14 

1507 

227 

.025     5637 

.28344 

.857 

1416 

.976  7805 

.19986 

706 

0.1s 

1734 

7,61 

0.027  49 '6 

4.28670 

1.863 

1622 

220 

9.975  2011 

4.1  971  2„             I 

700 

.16 

J977 

.029  4340 

.28999 

.869 

1842 

233 
2a6 

•973  6316 

.19440 

695     : 

•'7 

2238 

177 
29J 
311 

.031    3913 

.29331 

iP 

2075 

■972  0719 

.19170 

690       ; 

.18 

2515 

.033   3636 

.29665 

.882 

2321      260 

.970  5218 

.18901 

685 

.19 

2  8  09 

•°35  35" 

.30001 

.888 

2581     Z73 

.968  9813 

.18633 

679 

0.20 

3120 

3*8 
345 
363 
381 
398 

0.037  3542 

4^3°339 

1.895 

2854    286 

9.967  4502 

4.18367..             1 

672 

.21 

3448 

.039  3730 

.30679 

.901 

3  HO    299 
3439    ,,^ 

.965  9285 

.18102 

666         : 

.22 

3793 

.041  4077 

.31022 

.908 

.964  4159 

.17840 

661     ' 

•*3 

4156 

•043  4585 

.31368 

.915 

375' 

325 
338 

.962  9124 

•«7579 

b55 

;  -4 

45  37 

.045  5259 

.31716 

.922 

4076 

.961   4180 

.17319 

649     1 

i  0.25 

4935 

416 

434 
452 
471 
488 

0.047  6099 

4.32066 

1.929 

44'4 

1  C  I 

9.959  9324 

417061,         1 

643 

;    .2b 

5351 

.049  7109 

.32418 

.936 

VAl  363 

5'28       376 

5504     389 

.958  4556 

.16803 

637 

■-1 

5785 

.051   8290 

■32773 

•943 

.956  9875 

.16547 

t\3' 

i      •*«! 

6237 

.053  9646 

.33131 

.951 

•955  5281 

.16292 

625 

1      -'^ 

6708 

.056  1179 

.33492 

.958 

•954  0771 

.16038 

618 

;  °^3o| 

7196 

0.058  2893 

4.33856 

1.966 

6294 

9.952  6346 

4^1 5785,,        J 

6.3 

i 

TAP 

T.T!   Y 

Par+   T 

T 

, 

1 

T 

0.00 
.01 
.02 
.03 
.04 

0.05 
.06 

i     ■°l 
1     .08 

.09 

O.IO 

.11 

1        ''2 

i     •>3 

1  •H 

1   0.15 
.16 

1      -'7 

!   .18 

;     ..9 

1  0.20 

Ellipse. 

Ilypcrbolii. 

T 

Ellipse. 

Ilypeihola.           ! 

A 

Diir. 

A 

Dlff. 

A 

Diir. 

A 

Difl'. 

0.00000 
.0099a 
.01969 
.02930 
.03877 

0.04808 
.05726 
.06630 
.07521 
.08398 

0.09263 
..0I16 
.10956 
.11783 
.12599 

0.13404 
.14198 
.14981 

•>5753 
.16515 

0.17266 

992 

977 
961 

947 
931 

918 

904 
891 

877 
865 

^53 
840 

827 

816 

805 

794 

783 
772 
762 
7S» 

0.00000 
.01008 
.02033 
.03074 
.04132 

0.05209 
.06303 
.07417 
.08550 
.09702 

0.10875 
.12069 
.13285 
.14522 
.15782 

0,17067 
.18375 
.19709 
.21068 
.22454 

0.23867 

_ 

1008 

1025 
1041 
1058 

1077 

1094 
1114 
1133 
1152 
1173 

1194 
1216 

1237 
1260 
1285 

1308 
•334 
1359 
1386 
1413 

0.20 
.21 

.22 
•23 
•24 

0.25 
.26 

•27 
.28 
.29 

0.30 

•3' 
•32 
•33 
•34 

•39 
0.40 

0.17266 
.18008 
.18740 
.19462 

.20174 

0.20878 
.21573 
.22258 

•22935 
.23604 

0.24265 
.24917 
.25561 
.26198 
.26826 

0.27447 
.2S061 
.28668 
.29268 
.29860 

0.30446 

742 
732 
722 

;;: 
tn 

677 
669 
661 

652 
644 
637 
628 
621 

614 

607 
600 

III 

0.23867 
.25309 
.26779 
.28280 
.29813 

0.31377 

1442 

1470 
150I 

1564 

i 
1 
1 

i 

618 


rboHc 

Orbits. 

"~^      ^ 

1 
UifT.   , 

log 
mlfU.Diff. 

82,, 

1.771 

86 

.767 

J92 

.762 

398 

■758 

io7 

•753 

qi8„ 

..748 

130  1 

•743 

HI 

•739 

659 

•734 

37C 

.730 

o94„ 

1.725 

8.,- 

.720 

s-;7 

.716 

200 

.711 

986 

.706 

I712n 

1.700 

440 

.695 

(lyo 

.690 

(901 

.685 

5633 

.679 

^367,. 

1.672 

.666 

7840 

.661 

7579 

■655 

7319 

.649 

706 1  „ 

1.643 

6803 

.637 

6S47 

.631 

6292 

•5-3 

6038 

.618 

5785n 

1.613 

eiliola. 


Diff. 


1442 

1470 
I  501 

1533 
1564 


TABLE  XL 

For  tlie  Motion  in  n  Parabolic  Obit. 


V 

I"i?ft 

I)ilT. 

1 

0.000 

0.000  0000 

0.060 

.001 

.000  0000 

0 

.061 

.002 

.000  0001 

I 

.062 

.003 

.000  0002 

I 

.063 

.004 

.000  0003 

I 
I 

.064 

0.005 

0.000  0004 

0.065 

.006 

.000  0006 

2 

.066 

.007 

.000  0009 

3 

.067 

.008 

.000  0012 

3 

.068 

.009 

.000  0015 

3 
3 

.069 

0.010 

0.000  0018 

0.070 

.01 1 

.000  0022 

4 

.071 

.012 

.000  ooi6 

4 

.072 

.013 

.000  0031 

S 

.073 

.0,4 

.000  0035 

t 

.074. 

0.015 

0.000  0041 

0.075 

.016 

.000  0046 

I 

7 

8 
8 
8 
8 
9 

.076 

.017 

.000  0052 

.077 

.018 

.000  0059 

.078 

.019 

.000  0065 

.079 

0.020 

0.000  0072 

0.080 

.021 

.000  0080 

.081 

.022 

.000  0088 

.082 

.023 

.000  0096 

.083 

.024 

.000  0104 

.084 

;  0.025 

0.000  01 1 3 

0.085 

1  .026 

.000  0122 

9 

.086 

!  .027 

.000  0132 

10 

.087 

i  .028 

.000  0142 

10 

.088 

i  •°^9 

.000  0152 

lO 

II 

.089 

0.030 

0.000  0163 

II 
II 
12 
12 
'3 

0.090 

i  •°3i 

.000  0174 

.091 

i  •°'i- 

.000  0185 

.092 

•033 

.000  0197 

•093 

.034 

.000  02og 

.094 

0.035 

0.000  0222 

0.095 

.036 

.000  0235 

'3 
13 

.096 

.037 

.000  0248 

.097 

.038 

.000  0262 

'4 
'3 
'S 

14 
16 

•5 
16 

16 

.098 

.039 

.000  0275 

.099 

0.040 

0.000  0290 

O.IOO 

.041 

.000  0304 

.101 

.042 

.000  0320 

.102 

.043 

.000  0335 

.103 

.044 

,000  0351 

.104 

0.045 

0.000  0367 

16 
«7 
>7 
18 
18 

0.105 

.046 

.000  0383 

.106 

.047 

.000  0400 

.107 

.048 

.000  0417 

.108 

.049 

.000  0435 

.109 

1  0.050 

0.000  0453 

18 
«9 
19 
19 

20 

0.1  10 

.051 

,000  0471 

.III 

.052 
.053 

.000  0490 
.000  0509 

.112 
.113 

1  •°54 

1 

,000  01,28 

.114 

0.055 

0.000  0548 

0.1 15 

.056 

.000  0568 

20 

.116 

I  -057 

.000  0589 

2! 

21 

.117 

.058 

.000  0610 

.118 

.059 

.000  0631 

21 
21 

.119 

0.060 

0.000  0652 

0.120 

lot; /It 


065a 
0674 
0697 
0719 
0742 

0766 
0790 
0814 
0838 
0863 

0888 
0914 
0940 
0966 
0993 

1020 
1047 
1075 
1103 
II32 

1161 
1 190 
1219 
1249 

1280 

I311 
1342 

1373 
1405 

1437 

1470 

1502 

1536 
1569 
1603 

1638 
1673 
1708 

"743 
1779 

1815 
1852 
1889 
1926 
1964 

2002 
2040 
2079 

21l8 

2158 

2198 
2238 
2279 
2320 
2361 

2403 

1445 
2487 
2530 
2S73 
0.000  2617 


0.000 
.000 
.000 
.000 
.000 

0.000 
.000 
.000 
.000 
.000 

0.000 
.oco 
.000 
.000 

.000 

0.000 
.000 
.000 
.000 
.000 

0.000 
.000 

.000. 

.000 

.000 

0.000 
.000 
.000 

.000 

.000 
0.000 

.000 

.000 
.000 
.000 

0.000 
.000 

.000 
.000 
.000 

0.000 

.000 
.000 

.000 

.000 

0.000 
.000 
.000 

.000 

.000 
0.000 

.000 

.000 

.000 
.000 

0.000 
.000 

.000 

.000 

.000 


Ditl. 

1 

1 

0.120 

22 

.121  1 

23 

.122  i 

22 

.123  1 

*3 

.124 

*4 

^  1 

0.125  ; 

H 

.126 

24 

.127 

24 

.128 

2.S 
25 

.129  i 

26 

0.130 ! 

26 

.131  : 

26 

.132  ; 

27 
27 

•'33  i 
•>34 

27 
28 

0.135 ; 
.136 

28 
29 
29 

.139 1 

;   29 

0.140  j 
.141 

29 
30 
3' 
31 

.142 

■«43 
.144 

0.145 

31 
3J 

.146 

•«47 

32 
32 
33 

.148 
.149 

0.150 

32 

.151 

34 

.152 
•'53 
•154 

33 
34 

35 

0.155  ' 

35 
35 

36 

.156 

.158 
.159 

0.160 

37 

.161 

37 

.162 

37 
3« 
38 

.164 

38 

0.165 
.166 

39 

.167 

39 

.168 

40 
40 

.169  ' 

0.170 

40 

.171 

4« 

.172 

4« 
4' 
42 

•173 
.174 

0.175 

42 
42 

.176 
•177 

43 

.178 

1  43 

.179 

44 

0.180 

In- 


0.000  2617 
.000  2661 
.000  2705 
.000  2750 
.000  ^79 5 

0.000  2841 
.000  2886 
.000  2933 
.000  29^9 
.000  3026 


3074 
3121 
3169 
3218 
3267 

3316 
3365 
34'5 
3466 
3516 

3567 
3619 
3671 
3723 

3775 
3828 
3S82 

39  35 
39S9 

4044 

0.000  4099 
.000  4154 

.C  ;.-.  V 
.0  ■'  :; 
.001^  .1 


0.000 

.000 

.000 
.000 
.000 

0.000 

.000 

.000 
.000 

.000 
0.000 

.000 

.000 
.000 

.000 

0.000 

.000 

.000 

.000 
.000 


437S 

4435 
4493 
4551 
4609 

4668 
4726 
4786 
4846 
4906 

4966 

5027 
5088 

5150 
5212 


0.000  5274 

.000  5337 

.000  5400 

.000  5464 

.000  5528 

0.000  3592 

.000  5657 

.000  5722 

.000  5787 

.000  5853 

0.000  5919 


0.000 
.000 
.000 
.000 
.000 

0.000 
.000 
.000 
.000 
.000 

0.000 
.000 
.000 
.000 
.000 


Dlfl'. 


44 

44 

45 

46 

45 
47 
46 

47 
48 

48 
49 
49 
49 

49 
50 
5' 

50 
51 

52 
52 
52 
52 
53 

54 
53 

54 
55 
55 

55 
55 
56 

56 

5'^ 

5^ 
58 

59 

58 
60 
60 
60 
60 

61 
61 

62 
62 
62 

64 
64 
64 

65 
65 
65 
66 
66 


OIU 


TABLE  XI. 


For  the  Motion  in  a  I'anibolic  Orbit. 


V 

log  ii 

Diff. 

V 

ipg/i 

Diff. 

>? 

lOgfl 

1 

Diff. 

0.180 

0.000  5919 

67 
67 
67 

68 
68 

0.240 

0.001  0603 

no    ° 

300 

O.OOI  6733 

.181 

.000  5986 

.241 

.001  0693 

90 

301 

.001  6848 

"5 

.182 

.000  6053 

.242 

.001  0784 

9' 

91 

9» 

92 

302 

.001  6963 

"5 

116  1 

116  i 

117  ; 

.1X3 

.000  6120 

.243 

.001  0875 

303 

.001  7079 

.,84 

.000  6188 

.244 

.001  0966 

304 

.001  7195 

0.185 

0.000  6256 

69 
68 

0.245 

O.OOI  1058 

n»    ° 

305 

O.OOI  7312 

1 

.186 

.000  6325 

.246 

.001  1 1 50 

92 
92 

306 

.001  7429 

117 

117  i 

118  i 

.187 

.000  6393 

70 
69 

70 

.247 

.001  1242 

307 

.001  7546 

.188 

.000  6463 

.248 

.001  1335 

93 

308 

.001  7664 

.189 

.000  6532 

.249 

.001  1429 

94 
93 

309 

.001  7783 

119 
118  ' 

0.190 

0.000  6602 

0.250 

O.OOI  1522 

° 

310 

O.OOI  7901 

.191 

1    -191 
.193 

.000  6673 
.000  6744 
.000  6815 

71 

7' 
71 

■ill 

.253 

.001  1617 
.001  1711 
.001  1806 

93 

94 
95 

3«« 
312 

313 

.001  8020 
.001  8140 
.001  8260 

119 

120  ; 
lao  ; 

.194 

.000  6887 

72 
71 

.254 

.001  1901 

95 
96 

3'4 

.001  838: 

121  1 
121  [ 

0.195 

0.000  6959 

71 
73 

0.255 

O.OOI  1997 

96 

97 

315 

O.OOI  8502 

121  1 

122 

.196 

.197 

.000  7031 
.000  7104 

.256 

.257 

.001  2093 
.001  2190 

316 

317 

.001  8623 
.001  8745 

.198 

.000  7177 

73 

.258 

.001  2287 

97 

3.8 

.001  8867 

122 

.199 

.000  7250 

73 
74 

.259 

.001  2384 

11 

319 

.001  8989 

1  :.^2  ! 
124  i 

0.200 

0.000  7324 

0.260 

0.001  2482 

93   ° 
99 

320 

O.OOI  9113 

^  1 

.201 
.202 

.000  7399 
.000  74/3 

75 
74 

.261 
.262 

.001  2580 
.001  2679 

321 
322 

.001  9236 
.001  9360 

123  1 

124  j 

.203 

.000  7548 

75 
76 
76 

.263 

.001  2778 

99 

323 

.001  9484 

124  ' 

.204 

.000  7624 

.264 

.001  2877 

99 

100 

3M 

.001  g'  39 

125 

125  1 

0.205 

0.000  7700 

76 
77 
77 
77 
78 

0.265 

O.OOI  2977 

0 
100 

lOI 

lOI 
102 
lOI 

315 

0.001  97  34 

126  i 

126  ; 

127  1 
127 
127 

.206 
.207 
.208 
.209 

.000  7776 
.oco  7853 
.000  7930 
.000  8007 

.266 
.267 
.268 
.269 

.001  3077 
.001  3178 
.001  3279 
.001  33S1 

326 
327 
328 
329 

.001  9860 
.00:  9986 
.001  01 1  3 
.002  0240 

1  0.210 
.211 

0.000  8085 
.000  8163 

78 
79 
79 
79 
80 

0.270 
.271 

O.OOI  3482 
.001  3^85 

103   ° 

103 

103 

33° 
331 

0.002  0367 
.002  0495 

128  i 

.212 
.213 

.oco  8242 
.000  8321 

.272 
.273 

.001  3688 

.001  3791 

331 
333 

.002  0624 
.002  0752 

1 29  i 
128 

.214 

.000  8400 

.274 

.001  3894 

104 

334 

.002  0882 

130  ; 
129 

0.215 

0.000  8480 

80 
81 
81 
81 
82 

0.27s 

O.OOI  3998 

105   ° 

104 

106 

in  " 

335 

0.002  lOII 

.216 

.000  8560 

.276 

.001  4103 

336 

.002  1141 

1 30   1 

.217 

.000  8641 

•177 

.001  4207 

337 

.002  1272 

131  i 

131  ! 

131 

132  ■ 

.218 

.000  8722 

.278 

.001  4313 

338 

.002  1403 

.219 

.000  8803 

.279 

.001  4418 

1O3 
106 

339 

.002  1534 

0.220 

0.000  8885 

82 
82 

0.280 

O.OOI  4524 

in-,     ° 

340 

0.002  1666 

.221 
.222 

.000  8967 
.000  9050 

.281 
.282 

.001  4631 
.001  4738 

107 
107 

341 
342 

.002  1799 
.002  1931 

133 

132  : 

.223 

.000  9132 

.283 

.001  4845 

107 
108 
108 

U3 

.002  2065 

'34 

.224 

.000  9216 

.284 

.001  4953 

344 

.002  2198 

133 

»35 

0.225 

0.000  9300 

84 

85 
86 

0.285 

0.001  5061 

108 

345 

0.002  2333 

1 

.226 

.000  9384 

.286 

.001  i^i69 

346 

.002  2467 

'34  1 

.227 

.000  9468 

.287 

.001  5278 

1  09 
IIO 

log 
lli 

347 

.002  2602 

"35 
136 
136 
136 

137 

.228 

.000  9551 

.000  9638 

.288 

.001  5388 

348 

.002  2738 

.229 

.289 

.001  5497 

349 

.002  2874 

0.230 
.231 

0.000  9724 
.000  9810 

86 

87 
87 
87 
88 

0.290 
.291 

O.OOI  5608 
.001  15718 

no    ° 
III 

112 
112 
112 

350 
351 

0.002  3010 
.002  3147 

.232 
•133 

.000  9897 
.000  9984 

.292 
.293 

.001  5829 
.001  5941 

35^ 
353 

.002  3284 
.002  3422 

'38 
'38 

139 

.234 

.001  0071 

.294 

.001  6053 

354 

.002  3560 

0.235 

o.ooi  0159 

88 
88 

89 
90 

0.295 

O.OOI  6165 

..3     ° 

113 

114 

114 

114 

355 

0.002  ■'  99 

.236 

.001  0247 

.296 

.001  6278 

356 

.002  3838 

139 

139 

.237 

.001  0335 

.297 

.001  6391 

357 

.002  3977 

.238 

.001  0424 

.298 

.001  6505 

358 

.002  4  I  17 

140 
'4' 
«4'  1 

.239 

.001  0513 

.299 

.001  6619 

359 

.002  4258 

0.240 

O.OOI  0603 



0.300 

O.OOI  6733 

0 

360 

o.oci  4399 

1 

02U 


DifT. 


I 
4 
3 

)i 

10 

^o 

)0 

!: 

Da 
'3 
V5 
57 
89 

13 

36 
60 
84 
39 

34 
60 

86 

13 

40 

67 

95 
2+ 

5  2 

182 

III 

4J 
72 
03 
34 
66 
99 
3> 

&■; 


|3  3 


I  IS 

116  I 

116 

117 

117 

117 

118 

\\l 

119 
120 
120 
121 
121 

121 

122 
122 

X'.\2 
124 

123 
124 
124 
125 
125 

126 
126 

127 
127 
127 

128 

129 


130 
129 

130 
131 
131 

131 
132 

133 

132 

134 

I     133 

:  '35 
134 

135 
136 

130 
136 

137 


i  1  1 

137 

*4  ! 

,38 

)0 

.38 

139 

>9 

n 

139 
139 

140  . 

141   ; 

>4»  ! 

19 

1 

TABLE  XI. 


For  the  Motion  in  a  Puraholic  Orbit. 


l0K>i 


I 


0.360 
.361 
.362 

•363 
.364 

0.36:; 
.366 

■367 
.368 
.369 

0.370 
•371  ; 
•371 
•373 
•374  I 

°-375  • 
•376  i 
•377  I 
•37^1 
•379  I 

0.3S0  I 
.38, 
.382 
.383 
.384 

0.385 
.386 
.387 
.3i,.- 
.389 

0.390 
.391 
.392 
•393 
•3'>+ 

0.395 
.396 

•397 
.398 

•399 

0.400 
.401 
.402 
.403 
.404 

0.405 
.406 
.407 
.408 
.409 

0.410 
.411 
.412  { 
.413 
.414 

0.415 
.416 

•4»7 
.418 
.419 


0.002  4399 
.002  4540 
.002  4682 
.002  4824 
.002  4967 

0.002  5110 

.002  5254 

.002  5398 

.002  5543 

.002  5688 

0.002  5834 

.002  5980 

.002  6126 

.002  6273 

,002  6421 

0.002  6568 
.002  6717 

.002  6866 
.002  7015 
.002  7165 


I)ilT. 


7315  i 
7466  j 
7617  I 

7769  I 
7921  j 

8073  I 

8226 

8380 

8534 
8689  I 

8844  I 
8999  I 
9155  ; 

93>'  ! 
9468 

9626 
9784 
994^  ! 
oioi  j 
0260  I 


0.002 
.002 
.002 
.002 
.002 

0.002 
.002 
.002 
.002 
.002 

0.002 
.002 
.002 
.002 
.002 

0.002 
.002 
.002 
.003 
.003 

0.003  °420  I 

.003  0580  I 
.003  0741  I 
,003  0903 
.003  1064 

0.003 
.003 
.003 
.003 
.003 

0.003 
.003 
.003 
.003 
.003 

0.003 
.003 
.003 
.003 
.003 


122; 
1389 
1553 

1716 

I88I 

i°45 
221 1 
2376 

2543 

2709 

2877 

3044 
3213 
3381 
355° 


4« 

41 
42 
43 
43 

44 
44 
45 

46 

46 
46 

47 
48 

47 

49 
49 
49 
5° 
5° 

5« 
SI 

5^ 
52 
52 

53 
54 
54 
55 
55 

55 
56 
56 
57 
S» 

S« 

s» 

59 
59 
60 

60 
61 
62 
61 

63 
62 
64 

64 

66 
65 
67 
66 
68 

67 
69 
68 
69 

70 


log/* 


DIfl'. 


0.420  ;  0.003  3720 


0.120 

,-•.2  I 
.^22 

4^3 
424 

J.25 
4.^6 
427 
428 
429 

430 
431 
43i 
433 
434 

435 
436 

437 
438 

439 

440 
441 
442 
443 
444 

445 
446 

447 
448 

449 

450 

451 
452 

45  3 
454 

455 
456 

457 
45S 
459 

460 
46, 
462 
463 
464 

ill 
467 
468 
469 

470 
471 
472 

173 
474 

475 
476 

477 
478 

479 

480 


0.003 

.003 
.003 
.003 
.003 

0.003 
.003 
.003 
.003 
.003 

0.003 
.003 
.003 
.003 
.003 

0.003 
.003 
.003 
.003 
.003 


3720 
3890 
4061 
4232 
4404 

4576 
4749 
4V2  3 
509b 

5271 

5445 
5621  , 

5797  ' 
5973  ' 
6150  I 

6327  I 
6505  j 
6683  I 
6862 
7042 


0.003  7222 
.003  7402 
.003  7583 
.003  7765 
.003  7947 

0.003  8130 
.003  8313 
.003  8496 
.003  8680 
.003  88(-5 

0.003  9050 
.003  9236 
.003  9422 
.003  9609 
.003  9797 

0.003  9984 
.004  0173 
.004  0362 
.004  0551 
.004  0741 

0.004  0932 

.004  1 1 23 

.004  1315 

.004  1507 

.004  1700 

0.004  1893 

.004  2087 

.004  2281 

.004  2476 

.004  2672 

0.004 
.004 
.004 
.004 

.004 

0.004 
.004 
.004 
.004 
.004 


2868 
3064 
3261 

3459 

3657 

35*56 
4°55 
4^55 
4456 

4657 

0.004  4858 


70 
71 
71 
72 
72 

73 
74 
73 
75 
74 
76 
76 
76 
77 
77 

78 
78 

79 
80 
80 

80 
81 

«2 
82 
83 

83 
83 
84 
85 
85 

86 
86 

87 
88 

87 

89 
89 

89 

90 

91 

91 

92 
92 
93 
93 

94 
94 
95 
96 
96 

96 

97 
98 

98 

99 

199 

200 
201 
201 
201 


login 


Diff. 


0.480 
.481 
.482 
.483 

•484 
0.485 
.486 
•487 
.488 

•489 

0.490 
.491 
.492 
■493 
•494 

0.495 
.496 

•497 
.498 

•499 

0.500 

•51 
.52 

•53 
•54 

0-55 
.56 

•57 
•58 
•59 
0.60 
.61 
.62 

.64 

0.65 
.66 
.67 
.68 
.69 

0.70 

•71 
.72 

•73 
•74 

0.75 
.76 

•77 
•78 
•79 
0.80 
.81 
.82 

i^ 
.84 

0.85 

.86 

■87 
.88 
.89 

0.90 


0.004 

.004 
.004 
.004 
.004 

0.004 
.004 
.004 
.004 
.004 

0.004 
.004 
.004 
.004 
.004 

0.004 
.004 
.004 
.004 
.004 

0.004 
.005 
.005 
.005 
.005 

0.006 
.006 
.006 
.006 
.007 

0.007 

.007 
.007 
.008 
.008 

0.008 
.009 
.009 
.001; 
.010 


4858 
5061 

5263 

5467 
5670 

5875 
6080 

6285 
6492 
6698 

6906 
7113 

7322 
7531 
774° 

7951 
8161 

8373 
8585 
8797 

9010 
1173 

3397 
5681 
8029 

0441 
2919 

54fH 
8079 
0765 

35^5 
6361 

9^74 
2268 

5345 
8508 
1759 
5103 
8542 
2081 


I 


o.oio  5723 

.010  9473 

.on  3336 

.0.   !  7316 

.c-  .  1419 

0.012  5652 

.013  0022 

.013  4536 

.013  9202 

.014  4031 

0.014  9033 

.015  4219 

.015  9603 

.016  5202 

.017  1033 

0.017  7120 

.018  3486 

.019  0165 

•019  7195 

.020  4629 

0.021  2519 


203 

202 
204  j 
203 
205 


205 
207 
206 
208 

207 
209 
209 
209 
21  I 

210 
212 
212 
212 
213 

2163 
2224 

1284 
2348 
2412 

2478 

2545 
2615 
2686 
2760 

2836 
2913 
2994 

3077 
3163 

3251 

3  344 
3439 

3539 
3642 

3750 
3863 
3980 
4103 

4233 

4370 
4514 
4666 

4829 

5002 

5186 
5384 
5599 
5831 
6087 

6366 
6679 

7030 

7434 
7900 


621 


TABLE  XII. 


'< 

log  Jll] 

log  m-i 

^1' 

1 

z 

3 

1 

m 

ill. 

0  / 

mi 

m 

I 

m 

' 

)"a 

I 

o   / 

0 

/ 

0  / 

0  > 

0 

1 

0 

/ 

0   ' 

1 

0   -  1 

—  0  0 

00 

0.0000 

0 

0 

90  0 

90  0 

180  0 

180 

0 

180 

0 

0  0 

0  o| 

1 

4.2976 

9.9999 

-» 

13 

90  20 

90  20 

178  40 

178 

40 

'79 

0  359  0 

359  5' 

i) 

1   s 

3-395° 

9.9996 

4 

46 

90  40 

90  40 

177  20 

'77 

20 

178 

0  358  0 

358  9 

•] 

i.8675 

9.9992 

7 

8 

91  0 

91  0 

176  0 

176 

0 

'77 

0 

357  0 

357  '4 

4 

2.493» 

9.9986 

9 

3* 

91  20 

91  20 

'74  40 

'74  40 

176 

0 

356  0 

356  18 

m 

2.2044 

9.9978 

11 

55 

91  41 

91  41 

'73  '9 

173 

.9 

I7S 

0 

355  0 

355  23 

0 

1.96S6 

9.9968 

'4 

'9 

92  t 

92  I 

'7'  59 

'7' 

59 

174 

0 

354  0 

354  :^8 

7 

1.7698 

9-9957 

16 

4i 

92  •...'. 

92  22 

170  38 

170 

38 

172 

59 

353  ' 

353  32 

H 

1.5981 

9-9943 

'9 

7 

92  42 

92  42 

169  18 

169 

18 

171 

59 

352  I 

352  37 

9 

'•4473 

9.9928 

21 

3* 

93  3 

93  3 

167  57 

167 

57 

170 

58 

35'  2 

35«  42 

10 

1.3130 

9.991 1 

23 

57 

93  *S 

93  as 

166  35 

166 

35 

169 

57 

350  3 

35°  47 

11 

1. 1922 

9.9892 

26 

^3 

93  46 

93  46 

165  14 

165 

14 

168 

55 

349  4 

349  52 

12 

1.0824 

9.9871 

28 

50 

94  8 

94  8 

163  52 

'63 

5* 

167 

54 

348  6 

348  56 

13 

0.9821 

9.9848 

3' 

17 

94  3« 

94  3' 

162  29 

162 

29 

166 

S' 

347  8 

348  I 

14 

0.8898 

9.9823 

33 

46 

94  53 

94  S3 

161  7 

161 

7 

165  48 

346  II 

347  6 

15 

0.8045 

9.9796 

36 

'5 

95  17 

95  '7 

'59  43 

'59 

43 

164 

44 

345  '4 

346  II 

1(( 

0.7254 

9.9767 

38  46 

95  40 

95  40 

158  20 

,58 

20 

163 

40 

344  '7 

345  '6 

17 

0.6518 

9.9736 

4' 

18 

96  5 

96  5 

'56  55 

156 

55 

162 

34 

343  =' 

344  -I 

IS 

0.5830 

9.9702 

43 

5' 

96  30 

96  30 

1 55  30 

'55 

30 

161 

27 

342  27 

343  27 

10 

0.5185 

9.9667 

46 

26 

96  56 

96  56 

'54  4 

'54 

4 

160 

'9 

34'  r- 

342  32 

20 

0.45S1 

9.9629 

49 

^ 

97  23 

97  i3 

'5*  37 

15a 

37 

'59 

9 

340  38 

34'  37 

21 

0.4013 

9.9588 

5' 

41 

97  5° 

97  5° 

151  10 

'5' 

10 

'57 

58 

339  45 

34°  43 

1  o«> 

0.3479 

9.9545 

54 

22 

98  19 

98  19 

149  41 

'49 

4' 

156 

45 

338  53 

339  49 

1  23 

0.2976 

9.9499 

57 

5 

98  49 

98  49 

148  II 

148 

II 

'55 

29 

338  0 

338  54 

24 

0.2501 

9-945' 

59 

5' 

99  20 

99  20 

146  40 

146 

40 

'54 

II 

337  9 

338  0 

25 

0.2053 

9.9400 

62 

40 

99  53 

99  53 

'45  7 

145 

7 

152 

50 

336  19 

337  6 

2(> 

0.1631 

9-9345 

65 

33 

100  28 

100  28 

'43  32 

'43 

r- 

'5' 

25 

335  18 

336  '3; 

V. 

0.1232 

9.9287 

68 

30 

101  5 

loi  5 

'4'  55 

141 

55 

'49 

56 

334  38 

335  »9 

2S 

0.0857 

9.9226 

7' 

33 

loi  45 

loi  45 

140  15 

140 

■5 

148 

22 

333  49 

334  25: 

20 

0.0503 

9.9161 

74 

4' 

102  27 

102  27 

'38  33 

'38 

33 

146 

42 

333  ' 

333  32 

30 

0.0170 

9.9092 

77 

58 

103  13 

103  13 

136  46 

136  46 

'44 

55 

332  12 

332  39 

31 

9.9857 

9.9019 

81 

23 

104  4 

104  4 

134  56 

134 

56 

142 

59 

33'  24 

33'  46, 

32 

9.9^65 

9.8940 

85 

0 

105   I 

105  1 

132  59 

132 

59 

140 

5' 

330  37 

330  54! 

33 

9.9292 

9.8856 

88 

54 

106  6 

106  6 

'3°  54 

130 

54 

'38 

27 

329  49 

330   2| 

34 

9.9040 

9.8765 

93 

11 

107  22 

107  22 

128  38 

128 

38 

'35 

39 

329  2 

329  10 

35 

9.S808 

9.8665 

98 

7 

108  58 

108  58 

126  2 

126 

2 

132 

'3 

328  14 

328  19 

30 

9.8600 

9-8555 

104 

20 

III  13 

III  13 

122  47 

122 

47 

127 

29 

r-7  27 

327  28 

!— 3«  o2.2 

1 

9.8443 

9.8443 

116 

34 



116  34 

116  34 

"6  34 

116 

34 

116 

34 

326  45 

326  45 

This  tabic  exhibits  tlie  limits  of  the  roots  of  the  equation 

sin  (s'  —  C)  =  vio  sin*  z', 

when  tlierc  are  four  real  roots.  The  quantities  irii  and  ma  are  the  limiting 
values  of  m^,  and  the  values  of  2/,  z^',  z^',  and  z^',  corresponding  to  each  of 
these,  give  the  limits  of  the  four  real  roots  of  the  equation. 


622 


: 

1 

i 

i 

/ 

O    '   ' 

o 

o  o; 

o 

3^9     5' 

TABLE  XII. 


3  5«  9 
357  '4 
356   iS 

355  23 
354  ^S 
353  3^ 
35i  37 
35«  41 


35°  47 
349  5- 
348  56 

r   8  I  348   I 

ill  347  6 

;  Hi  346  •! 

^  »7j  345  '6 
5  21  I  344  -I 

1  27  343  -7 
I    32 1  34a  31 


38 

45 

53 

o 

9 


34»  37 
340  43 
339  49 
338  54 


b  19 

5  i« 
4  38 
3  49 
3  ' 

2  12 

I  24 
1°  37 
9  49 


338 

0 

337 
336 

6 
>3 

335 

19 

334 

15; 

333 

J" 

33* 

33> 

39 

46 

330 

54' 

i  330 

•*    r 

9  i 

329  10 

8  14 

328  19 

-  27 

327  i8 

6  45 

326  45 

C 

log  m. 

log  m.j 

i 

1 

z 

J 

r 

»' 

\                             \ 

"a 

n 

" 

"'. 

nig 

™, 

"'1 

'"i 

m^ 

0  » 

0  ' 

0 

1 

0  1 

0  1 

0   ' 

0  / 

0  ' 

0   / 

+  00 

00 

0.0000 

0  0 

0 

0 

0  0 

90  0 

90  0 

180  0 

180  0 

180  0 

1 

4.2976 

9.9999 

I  0 

I 

20 

I  20 

89  40 

89  40 

177  37 

180  55 

181   0 

2 

3-3950 

9.9996 

a  0 

2 

40 

2  40 

89  20 

89  20 

175  «4 

181  51 

182   Ol 

» 

2.8675 

9.9992 

3  0 

4 

0 

4  0 

89   0 

89  0 

172  52 

1S2  46 

183  0 

4 

2.4938 

9.9986 

4  0 

5 

20 

520 

88  40 

88  40 

170  28 

183  42 

184  0 

A 

2.2044 

9.9978 

S  ° 

6 

41 

6  41 

88  19 

88  19 

168  s 

184  37 

185  0 

1   t* 

1.9686 

9.9968 

6  0 

8 

I 

8  I 

87  59 

87  59 

165  41 

185  32 

186  0 

1   7 

1.7698 

9-9957 

7  I 

9 

22 

9  22 

87  38 

87  38 

163  18 

186  28 

.86  59 

H 

1. 5981 

9-9943 

8  I 

10 

42 

10  42 

87  18 

87  18 

160  S3 

1S7  23 

187  59 

9 

'•4473 

9.9928 

9  a 

12 

3 

12  3 

86  57 

86  57 

158  28 

188  18 

188  58 

10 

1. 3130 

9.9911 

10  3 

>3 

25 

13  2; 

86  35 

86  35 

156  3 

189  13 

189  57 

11 

i.:922 

9.9892 

J«  5 

J4 

46 

14  46 

86  14 

86  14 

«53  3/ 

190  8 

190  56 

12 

1.0824 

9-987« 

12   6 

16 

8 

i6  8 

85  52 

85  52 

151  10  .-91  4 

191  54 

13 

0.9821 

9.9848 

13  9 

17 

31 

17  3' 

85  29 

8s  29 

148  43  9'  59 

192  52 

14 

0.8898 

9.9823 

14  12 

18 

53 

18  53 

85  7 

85  7 

146  14  192  54 

'93  49 

15 

0.8045 

9.9796 

15  16 

20 

"7 

20  17 

84  43 

84  43 

'43  45  '93  49 

'94  46 

IC 

0.7254 

9.9767 

16  20 

21 

40 

21  40 

84  20 

84  20 

141  14 

1:4  44 

195  43 

17 

0.6518 

9.9736 

17  26 

23 

5 

23  5 

83  55 

83  SS 

138  42 

'95  39 

-.6  39' 

IS 

0.5830 

9.9702 

18  33 

44 

30 

•H  30 

83  30 

83  30 

136  9 

196  33 

197  33 

lU 

0.5185 

9.9667 

19  41 

25 

56 

25  56 

83  4 

83  4 

133  34  197  28 

198  28 

20 

0.4581 

9.9629 

20  51 

27 

23 

27  23 

82  37 

82  37 

130  58  198  23 

'99  22 

21 

0.4013 

9-9588 

22  2 

28 

SO 

28  50 

82  10 

82  10 

128  19  i9c^  17 

200  1 5 , 

22 

0.3479 

9-9545 

a3  15 

30 

>9 

30  19 

81  41 

81  41 

125  38  200  II 

701  07 

23 

0.2976 

9-9499 

24  31 

3» 

49 

3'  49 

81  II 

81  II 

122  55 

201   6 

2-02   Oi 

24 

0.2501 

9-945 « 

25  49 

33 

20 

33  20 

80  40 

80  40 

120  9 

202  0 

202  5  I  \ 

25 

0.2053 

9.9400 

27  10 

34 

53 

34  53 

80  7 

80  7 

117  20 

202  54 

203  41; 

20 

0.1631 

9-9345 

a8  35 

36 

28 

36  28 

79  32 

79  32 

114  27 

203  47 

204  32 

27 

0.1232 

9.9287 

30  4 

38 

5 

38  5 

78  55 

78  55 

III  30  204  41 

205  22 

2.S 

0.0857 

9.9226 

3'  38 

39 

45 

39  45 

78  IS 

78  15 

108  27  205  35 

206  II 

20 

0.0503 

9.9161 

33  18 

41 

27 

41  27 

77  33 

77  33 

los  19 

206  28 

206  59 

30 

0.0170 

9.909a 

35  5 

43 

13 

43  J3 

76  47 

76  4; 

102  3 

207  21 

207  48 

31 

9.9857 

9.9019 

37  » 

45 

4 

45  4 

75  56 

75  56 

98  37 

208  14 

208  36 

32 

9-9565 

9.8940 

39  9 

47 

I 

47  > 

74  59 

74  59 

95  0 

209  06 

209  23 

33 

9.9292 

9.8856 

41  33 

49 

6 

49  6 

73  54 

73  54 

91  6 

209  58 

210  II 

34 

9.9040 

9-8765 

44  21 

5' 

22 

51  22 

72  38 

72  38 

86  49 

7  10  50 

210  58 

35 

9.8808 

9.8665 

47  47 

53 

58 

S3  58 

71  2 

71  2 

81  53 

211  41 

211  46 

3(t 

9.8600 

9-8555 

52  31 

57 

13 

57  13 

68  47 

68  47 

75  40 

212  32 

212  33 

-f-36  52.2 

9.8443 

9-8443 

63  26 

63 

26 

63  26 

63  26 

63  26 

63  26 

213  15 

213  «5, 

This  table  exhibits  the  limits  of  the  roots  of  the  equation 

sin  (3'  —  C)  =  «!o  sin*  z', 

^he  liiniting   I  when  there  are  four  real  roots.     The  quantities  r\\  and  ?%  are  the  limiting 
to  each  of    I  values  of  m^,  and  the  values  of  s/,  Sj',  s,',  and  r/,  coi'responding  to  each  of 
these,  give  the  limits  of  the  four  real  roots  of  the  equation. 


623 


TABLE  XIII. 

For  finding  tlie  Ratio  of  the  Sector  to  the  Trianjcle. 


1 

lot! 

s"- 

o.oooo 

0.000 

0000 

.OOOI 

,000 

0965 

.0002 

.000 

1930; 

.0003 

.000 

"■I'H 

.0004 

.000 

3858 

0.0005 

0.000 

4821  i 

.0006 

.000 

';7X4 

.0007 

.000 

6747 

.0008 

.000 

7710 

.0009 

.000 

8672 

O.OOIO 

0.000 

9634 

.0011 

.001 

0595 

.0012 

.001 

•  556 

.0013 

.001 

2517 

.0014 

.001 

347« 

0.0015 

O.OOI 

4438 

.0016 

.001 

5398 

.0017 

.001 

6357 

.0018 

.001 

7316 

.0019 

.001 

8275 

0.0020 

O.OOI 

9234 

.0021 

.002 

0192 

.0022 

.002 

1150 

.0023 

.002 

2107 

.0024 

.002 

3064 

0.0025 

0.002 

4021 

.0026 

.002 

4977 

.0027 

.002 

5933 

.0028 

.002 

6889 

.0029 

.002 

7845 

0.0030 

0.002 

8800 

.0031 

.002 

9755 

.0032 

.003 

0709 

.0033 

.003 

1663 

.0034 

.003 

2617 

0.0035 

0.003 

3570 

.0036 

.003 

4523 

.0037 

.003 

5476 

.0038 

.003 

6428 

.0039 

.003 

7380 

0.0040 

0.003 

8332 

.0041 

.003 

9284 

.0042 

.004 

°^J? 

.0043 

.004 

ii86 

.0044 

.004 

2136 

0.0045 

0.004 

3086 

.0046 

.004 

4036 

.0047 

.004 

4985 

.0048 

.004 

5934 
6883 

.0049 

.004 

0,0050 

0.004 

iH^ 

.0051 

.004 

8780 

.0052 

.004 

9728 

.0053 

.005 

0675 

.0054 

.005 

1622 

0.0055 

0.005 

2569 

.0056 

.005 

35'5 

.0057 

.005 

4461 

.0058 

.005 

54°7 

.0059 

.005 

6353 

0.0060 

0.005 

7298 

965 
965 
964 
964 
963 

963 
963 
963 

962 
962 

g6i 
961 
961 
961 
960 

960 
959 
959 
959 
959 

958 
958 
957 
957 
957 

956 
956 
956 
956 
955 

955 
954 
954 
954 
953 

95  3 
953 
952 
952 
952 

952 
951 
95> 

950 
950 

95° 
949 
949 
949 
949 

948 
948 
947 
947 
947 

946 
946 
946 
946 
945 


logs3  I    DitT. 


0.0060 

.0061  : 

.0062' 
.0063 
.0064 

o.ocfi5 
.0066 
.0067 

.0068 

.0069 

0.0070 
.0071 
.0072 

.0073 
.0074 

0.0075 
.0076 
.0077 
.0078 
.0079 

0.0080 
.0081 
.0082 
.0083 

.00S4 

0.0085 

.0086 

.0087 

.0088 

.0089 

0.0090 
.0091 
.0092 
.0093 

.0094 
0.0095 

.0096 

.0097 

.0098 
.0099 

O.OIOO 

.0101 

.0102 
.0103 
.0104 

0.0105 
.0106 
.0107 
.0108 
.0109 

o.oi  10 
.0111 
.0112 
.0113 
.0114 

0.0115 
.0116 

.0117 

.0118 

.0119 


0.005  7298 
.005  8243 
.005  9 1 87 
.006  0131 
.006  1075 


0.006 
.006 
.006 
.006 
.006 

0.006 
.006 
.006 
.006 
.007 

0.007 
.007 
.007 
.007 
.007 

0.007 
.007 
.007 
.007 
.007 

0.008 
.008 
.008 
.008 
.008 

0.008 
.008 
.008 
.008 
.008 

0.009 
.009 
.009 
.009 
.009 

0.009 

.009 
.009 
.009 
.009 


2019 
2962 
3905 

4X47 
5790 

6732 
7673 
8614 

9555 
0496 

1436 
2376 
3316 
4^55 
5 1 94 
6133 
7071 
8009 

8947 
9884 

0821 
1758 
2694 
3630 
4566 

5502 
6437 
7372 
8306 
9240 

0174 
1108 
2041 
2974 
3906 

4838 

5770 
6702 
7633 
8564 


0.009  9495 

.010  0425 

.010  1355 

.010  2285 

.010  3215 

o.oio  4144 

.010  5073 

.010  6001 

.010  6929 

.010  7857 

O.OIO  8785 

.oio  9712 

.Oil  0639 


.oil 
.oil 


1565 
2491 

3417 


945 
944 
944 
944 
944 

943 
943 
942 

943 
942 

941 
94' 
941 
94' 

940 

940 
940 

939 
939 
939 
938 
938 
938 
937 
937 

937 
936 
936 
936 
936 

935 
935 
934 
934 
934 

934 
93  3 
933 
932 
932 

932 
932 

931 
931 

931 

930 
93° 
93° 
930 
929 

929 
928 
928 
928 
928 

927 
927 
926 
926 
926 


log  «2 


ma: 


0.0120  0.0 

.0121c  .0 

.0122'  .0 

.0123J  .0 

.01241  .0 

'  1 


.0 


0.0125;  °-° 
.0126!     .0 

.OI27J  .0 
.01281  .0 
.0129'     .0 

0.0130^  0.0 
.0131,  .0 
.0132  .0 
.0133!  .0 
•01341     -o 

0.0135!  0.0 
.0136      .0 

.0137 
.0138 
.0139 

0.01401  0.0 
.0141!  .0 
.OI42J  ,0 
.0143     ■° 

.0144 

0.0145 
.0146 

.0147 
.0148 
.0149 

0.0150 
.0151 
.0152 

•0'53 
.0154 

0.0155 
.0156 
.0157 
.0158 
.0159 

0.0160 
.0161] 
.01621 
.0163' 
.0164S 

0.0165; 
.oi66| 
.0167; 
.oi68i 
.0169 


0.0170 
.0171 
.0172 
.0173 
.0174 

0.0175 
.0176 
.0177 
.0178 
.0179 

0.0180 


I 


0.0 
.0 
.0 
.0 
.0 

0.0 
.0 
.0 
.0 
.0 

0.0 
,0 
.0 
.0 

.0 

0.0 
.0 
.0 
.0 
.0 

0.0 
.0 
.0 
.0 
.0 

0.0 
.0 
.0 
.0 
.0 

0.0 
.0 
.0 
.0 
.0 

0.016 


3417 
4343 
5268 

6193 

7118 

8043 
8967 
9890 
0814 
'737 
2660 
3583 
4505 
5417 
6348 

7269 
8190 
9111 

0032 
0952 

1871 
2791 
3710 
4629 
5547 
6465 

7383 
8301 
9218 
0135 

1052 ! 
1968  { 
2884 1 
3800 1 
4716 1 

5631 
6546  j 

7460 

8374 
9288 

5  0202 

5 

5 

5 

5 

5 
5 

5 
5 


1115 

2028 
2941  I 
3854: 
4766 
56781 
6589  I 
7500' 
841 1  ! 

9322 
0232 
1142 
2052 
2961 

3870 

4779 
5688 
6596 
7504 
8412 


926 
925 
925 
925 
925 

924 
923 
924 
923 
923 

923 
922 
922 
921 
921 

121 

i;2l 
921 
920 
919 

920 
919 
919 
918 

918 

918 
918 
917 
917 
917 

916 
916 
916 
916 
9' 5 

9'5 
914 
914 
914 
914 

913 
913 
913 
913 
912 

912 
911 
911 
911 
911 

910 
910 
910 

909 
909 

909 
909 
908 
908 
908 


C24 


TABLE  XIII. 

For  finding  the  Ratio  of  the  Sector  to  the  Triangle. 


log  .a 


923 
922 
922 
921 
921 

121  1 
>t2I  i 
921  ' 
920 
919 

920 
919 
919 
918 
918 

918 
918 

9'7 
i  9>7 
1  917 

916 
916 
916 
916 
9«5 

915 
914 
914 
914 
914 

913 
913 
913 
913 
912 

912 
911 
911 
911 
911 

910 
910 
910 
909 
909 

909 

Tol 
908 
908 


0.0180 
.0181 
.0182 
.0183 
.0184 

0.0185 
.0186 
.0187 
.0188 
.0189 

0.0190 
.0191 
.0192 
.0193 
.0194 

0.0195 
.0196 
.0197 
.0198 
.0199 

0.0200 
.0201 
.0202 
.0203 
.0204 

0.0205 
.0206 
.0207 
.0208 
.0209 

0.0210 
.0211 
.0212 
.0213 
.0214 

0.0215 
.0216 
.0217 
.0218 
.0219 

0.0220 
.0221 
.0222 
.0223 
.0224 

0.0225 
.0226 
.0227 
.0228 
.0229 

0.0230 
3231 
.0232 
.0233 
.0234 

0.0235 
.0236 
.0237 
.0238 
.0239 

0.0240 


0.016  8412 
.016  9319 

.017  0226 
.017 1133 
.017  2039 

0.017  *94S 

.017  3851 

.017  4757 

.017  5662 

.017  6567 


0,017 
.017 
.017 
.018 
.018 

0.018 
.018 
.018 
.018 
.018 

0.018 
.018 
.018 
.018 
.019 


747 » 
8376 
9280 
0183 
1087 

1990 
2893 

3790 
4698 
5600 

6501 
7403 
8304 
9205 
0105 


0.019  1005 

.019  1905 

.019  2805 

•019  3704 

.019  4603 

0.019  55°^ 

.019  6401 

.019  7299 

.019  8197 

.019  9094 

0.019  9992 

.020  0889 

.020  1785 

.020  2682 

.020  3578 

0.020 
.020 
.020 
.020 
.020 


4474 
5369 
6264 

7159 
8054 

8948 
9842 
0736 
1630 
2523 

3416 
4309 
5201 
6093 
6985 

7876 
8768 
9659 
0549 
1440 

0.022  2330 


0.020 
.020 
.021 
.021 
.021 

0.021 
.021 
.021 
.02 1 
.021 

0.021 
.021 
.021 
.022 
.022 


niff. 


907 
907 
907 
906 
906 

906 
906 
905 
905 
904 

905 
904 

903 
904 
903 

903 
903 
902 
902 
901 

902 
901 
901 
900 
900 

900 
900 
899 
899 
899 

899. 
898 
898 
897 
898 

897 
896 
897 
896 
896 

895 
895 
895 
895 
89+ 

894 
894 
894 
893 
893 

893 
892 
892 
892 
891 

892 
891 
890 
891 
890 


logj5 


0.0240 
.0241 
.0242 

•0*43 
.0244 

0.0245 
.0246 
.0247 
.0248 
.0249 

0.0250 
.0151 
.0252 
.0253 
.0254 

0.0255 
.0256 
.0257 
.0258 
.0259 

0.0260 
.0261 
.0262 
.0263 
.0264 

0.0265 
.0266 
.0267 
.0268 
.0269 

0.0270 
.0271 
.0272 
.0273 
.0274 

0.0275 
.0276 
.0277 
.0278 
.0279 

0.0280 
.0281 
.0282 
.0283 
.0284 

0.0285 
.0286 
.0287 
.0288 
.0289 

0.0290 
.0291 
.0292 
.0293 
.0294 

0.0295 
.0296 
.0297 
.02981 
.0299! 

0.0300'  o 


0.022 
,022 
,022 
022 
,022 


022 
022 
022 
022 
023 

023 
023 
023 
023 
023 

023 
023 
023 
023 
023 

024 
024 
024 
024 
024 

024 
024 
024 
024 
024 

024 
024 
025 
025 
025 

025 
025 
025 
025 
025 

025 
025 
025 
026 
026 

026 
026 
026 
026 
026 

026 
026 
026 
026 
026 

027 
027 
027 
027 
027 

027 


2330 
3220 
4109 
4998 
5887 

6776 
7664 
8552 
9440 
0328 

1215 

2102 
2988 

387s 
4761 

5647 
6532 

74'7 
8302 
9187 

0071 
0956 
1839 
2723 
3606 

4489 
5372 
625A 
7136 
8018 

8900 
9781 
0662 

•543 
2423 

3303 
4183 
5063 

5941 
6821 

7700 
«S79 
9457 
0335 
1213 

2090 
2967 

3«44 
4721 

5597 

6473 
7349 
8224 
9099 
9974 
0849 

1713 
2597 
347 « 
4345 
5218 


Dicr. 


890 
8§9 
889 
889 
889 

888 
888 
888 
888 
887 

887 
886 
887 
886 
886 

885 
885 
885 
885 
884 

885 

883 
884 
883 
883 

883 
882 
882 
882 
882 

881 
881 
881 
880 
S80 

880 
880 
879 
879 
879 

879 

878 
878 
878 
877 

877 
877 
877 
876 
876 

876 

875 
875 
875 
875 

874 
874 

874 
874 
873 


Io«»5 


0.0300' 
.0301 
.0302 
.0303' 
.0304, 

0.0305' 
.03061 

•03071 
.0308, 
.0309 

0.0310 
.031 
.0312 

•03>3 
.0314 

0.0315 
.0316 
.0317 
.0318 
.0319 

0.0320 
.0321 
.0322 
.0323 
.0324 

0.0325 
.0326 
.0327 
.0328 
.0329 

0.0330 
.0331 
.0332 
.0333 
.0334 

0.0335 
.0336 
•0337 
•0338 
.0339 

0.0340 
.0341 
.0342 
•°343 
•0344 

0.0345 
.0346 
.0347 
.0348 
.0349 

0.0350 
.0351 
.0352 

•0353 
.0354 

0.0355 
.0356 

•0357 
.0358 
.0359 

0.0360 


0,027 
.027 
.027 
.027 
.027 

0.027 
.028 
.028 
.028 
.028 

0.028 
.028 
.028 
.028 
.028 

0.028 
.028 
.029 
.029 
.029 

0.029 
.029 
.029 
.029 
.029 

0.029 
.029 
.029 
.029 
.030 

0.030 
.030 
.030 
.030 
.030 

0.030 
.030 
.030 
.030 
.030 

0.030 
.031 
.031 
.031 
.031 

0.031 
.031 
.031 
.031 
.031 

0.031 
.031 
.032 
.032 
.032 

0.032 
.032 
.032 
•032 
.032 

0.035   7120 


DifT. 


5218 
6091 
6964 
7836 
8708 

9580 
0452 

>3i3 
2194 
3065 

3936 
4806 
5676 
6546 
74«5 
8284 

9«53 

0022 
0890 
1758 

2626 

3494 
4361 
5228 
6095 

6961 

7827 
8693 ! 

9559  ; 

04241 

1290  : 

2154  1 

3019    ! 

38831 
4747 


56m| 
6475 

7338 
8201 
9064 

9926 

0788  ; 

1650  ; 
2512  ' 

3373  . 

4234  1 

5095 

5956 

6816 

7676 

8536- 
9396 

0255 ; 
1114 

1973 1 

2831  ! 
3689! 

4547 
5405 
6262 


873 
873 

87a 
872 
S72 

872 
871 
871 
871 
871 

870 
870 
870 
869 
869 

869 
869 
868 
868 
868 

868 
867 
867 
867 
866 

866 
866 
866 
865 
866 

864 
865 
864 
864 
864 

864 
863 
863 
863 
862 

862 
862 
862 
861 
861 

861 

861 
860 
860 
860 

860 
859 
859 

85? 
858 

858 
858 
858 

857 
858 


40 


U25 


TABLE  XIII. 

For  findinp  tlie  Kiitio  of  ilie  tScctor  to  the  TrinnRlc. 


logja 


0.0360 
.0361 
.0362 
.0363 
.0364 

0.03651 
.0366 
.0367 
.0368 
.0369 

0.0370 
.0371 
.0371 

•0373 
.0374 

0.0375 
.0376 
.0377 
.0378 
.0379 

0.0380 
.0381 
.0382 
.0383 
.0384 

0.0385 
.0386 
.0387 
.0388 
.0389 

0.0390 
.0391 
.0392 
.0393 
.0394 

0.0395 
.0396 
.0397 
.0398 
.0399 

0.040 
.041 
.o.;2 
.043 

.044 1 

0.045 
.046 
.047 
.048 
.049 

0.050 
.051 
.052 
.053 
.054 

0.05s 
.056 
.057 
.058 
.059 

0.060 


0.032 

.032 


032 
033 

033 

°33 
033 
033 

033 
033 
°33 
033 

033 

033 

034 
034 
034 
034 

034 

034 

°34 

03+ 
034 

034 
034 

03  s 
035 
035 

035 
035 
03  s 
035 
035 


PifT. 


7120 
7976 

i<^33 
9689 

0546 

1401  j 

**57  I 
31 12  ; 

3967  i 

48221 

5677  I 
6531 

73«5i 

8239 

9092 

9946 
0799 
1651 
2504 
3356 

420S 
5059 

6762 
7613 

8464 

93«4 
0164 
1014 
1864 

2713 
3562 
441 1 

5^59 
6108 


03s  6956 
035  7804 
035  8651 

035  9499 

036  0346 

036  1192 

036  9646 

037  8075  I 

038  6478 

039  4856 

040  3209 

041  1537 

041  9041 

042  8121  ° 

043  6376  p^l 

044  4607  ■ 


856 

857 
856 

857 
855 
856 
855 
855 
85s 
855 

854 
854 

854 
853 
854 

852 

853 
852 
852 

85, 
852 

85, 

850 
850 
850 
850 
849 

849 
849 
848 
849 
848 

848 

847 
848 
847 
846 

8454 
8429 
8403 
8378 

8353 

8328 

8304 
8280 


045 


281J 


8207 
046  0997  \l\ll 
046  9»57'  '^° 


047  7294 

048  5407 


^'37 
!ii3 

8089 


049  3496  '^ 
052  5626  I 


0.060 
.061 
.062 
.063 
.064 

0.065 
.066 
.067 
.068 
.069 

0.070 

.071 
.072 

•073 
.074 

0.075 
.076 
.077 
.078 
.079 

0.080 
.081 
.082 
.083 
.084 

0.085 
.086 
.087 
.088 
.089 

0.090 
.091 
.092 
.093 
.094 

0.095 
.096 
.097 
.098 
.099 

0.1 00 
.101 
.102 
.103 
.104 

0.105 


.108 
.109 

O.IIO 


.114 

0.1 1 5 
.116 
.117 
.118 
.119 


logs!   I  Dinr. 


0.052 
.053 
.054 
.054 
.055 

0.056 
.057 
.058 
.058 
.059 

0.060 
.061 
.061 
.062 
.063 

0.064 
.065 
.065 
.066 
,067 

0.068 
.068 
.069 
.070 
.071 

0.071 
.072 
.073 
.074 
.074 

0.075 
.076 
.077 
.077 
.078 


5626  I  , 
3602  ''^76 
1556  7954 
9488  7932 

7107     7909 

7397   ^ijgg 

0994  7»44 
88.7  7823 
66,8|78oi 

7780 

4398  I 
2157  ^^^59 
989  |7738 
76 1 2  7717 

53°«  life 

2984  , 

0639  765s 

8274:7635 

5888:''6'4 

3482,7594 

^^   7575 

1057 

8612 


6146 
3661 
II 


7575 

7555 
75  34 
7515 


)6l   '3*3 

■57  ■^■^'^A 
^'   7476 

8633 

6000  !7457 

3527  m 

0945  ,7+'8 

8345  ^T 
^^■'    7380 


5725 
3087 
0430 

77S4 
5060 

0.079  2348 
.079  9617 


7362 

7343 
7324 
7306 
7288 

7269 


80  6868  7=51 


.081  4101 
.082  1316 

0.082  8513 

.083  5693 

.084  2854 

.084  9999 

.085  7125 


0.086 


4235 


.087  1327 
.087  8401 
.088  5459 
.089  2500 

0.089  9523 
.090  6530 
.091  3520 
.092  0494 
.092  7451 

0.093  4391 
.094  1315 
.094  8223 
.095  5114 
.096  1990 

0.096  8841 


7233 
7215 

7197 

7180 
7161 

7145 
7126 

7110 

7092 

7074 
7058 
7041 
7023 

7007 
6990 
6974 
6957 
6940 

6924 
6908 
6891 
6876 
6859 


0.120 
121 
,122 
123 
124 

»25 
126 

127 

128 

129 

130 
131 
132 
133 

134 

135 

136 

'37 
138 

139 

140 
141 
142 
143 
144 

146 

'47 
148 
149 

150 

«Si 
152 

153 

154 

"55 
156 

157 
158 

159 
i6c 
161 
162 
163 
164 

165 
166 
167 
168 
169 

170 

'71 

172 

'73 
'74 

175 
176 
177 
178 
179 


logi« 


0.096 

,097 
098 
098 
099 

100 

100 
lOI 
102 
102 

103 

104 
104 
105 
106 

106 
107 
108 
108 
109 

110 

I  10 

III 

112 
112 

113 
114 
114 

116 
116 

"7 
118 
118 
119 


Diff. 


8849 
5692 
2520 

933« 

6127 

2907 
9672  ; 
6421  I 
3'54; 
9873! 

6576  i 

326A; 

9936 

6594 

3237 

9865 
647  8 

9660 
6229 

2783 

9323 

5849 
2360 

8§S7 

5340 
1809 
8264 
4704 
1131 

7S44 
3943 
0329 
6701 
3059 


6843 
6828 
6S11 
6796 

6780 

6765 
'■'749 
6733 
6719 
6703 

6688 
6672 
6658 
6643 
6628 

6613 
6598 
6584 
6569  I 
655t 

6540 
6526 
6511 
6497 
6483 

6469 

645s 
6440 
6427 
6413 

6399 
6386 
6372 
6358 
6345 


33' 


119  9404  , 

!!?  5735  e^Jg 
6304 
6292 
6278 

6265 
6*252 


121  2053 

121  8357 

122  4649 

123  0927 

123  7192 

124  3A44 


124  9082 

125  5908 

126  2121 

126  8321 


ttt!6238 


6226 
'3 


62 
I  62 


6187 


127  4508  V 

128  0683  I  ^'75 

128  6845  '"''^ 

129  2994 
129  913'  i  6 


«30  5255  I  6 
131  1367 


6165 
149 

6137 
124 


131  7466 

132  3553  i 

132  9628  I 

133  5690, 

134  1740 


60Q 
608 


6075 
6062 
6050 
6038 


347778:^^30 
135  3804 


we 


TABLE  Xm. 

For  fincling  the  Riitio  of  tla-  Sector  to  the  Triangle. 


1"K  »« 


Did'. 


6^.6 
65..  1 

6497  r 
6483 

6469 

6455! 
6440 , 

6427 
6413 

6399 

6386 1 

I  6371 : 

163581 

!  6345  j 

63311 
6318 
6304 
6292  i 
6278 1 

6265  j 
6252 
1 6238 
I  6226 

16213! 

i  6200 
i  6187 

I  6175 
1 6162 

1 6149 

1 6137 
6124 
6112 
6099 
6087 

6075 
6062 
6050 
6038 
6026 


0.180 

.l8i 
.182 
.i8j 
.184 

0.181; 
.186 
.187 
.188 
.189 

0.190 
.191 
.192 
.193 
.194 

0.195 
.196 
.197 
.198 
.199 

0.200 
.201 
.202 
.203 
.204 

0.205 
.206 
.207 
.208 
.209 

o.iio 
.211 
.212 
.213 

.214 

0.215 
.216 
.217 
.218 
.219 

O.220 
.221 
.222 
.223 
.224 

0.225 
.226 
.227 
.228 
.229 

0.230 
.231 
.232 

.234 

0.23s 
.236 

•237 
.238 
.239 

0.240 


0.135 

.136 
•«37 
•«37 
0.138 
.138 
.139 
.140 
.140 

0.141 
.141 

.142 

.143 

•«43 

0.144 

•'44 
.145 
.146 
.146 

0.147 
.147 
.148 
.148 
.149 

0.150 
.150 
.151 
.151 
.152 

0.152 
•«53 
•'54 
•'54 
•'55 

0.155 
.156 
.156 
•157 
•'57 

0.158 
•'59 
•'59 
.160 
.160 

0.161 
.161 
.162 
.162 
.163 

0.164 
.164 
.165 
.165 
.166 

0.166 
.167 
.167 
.168 
.168 


3^i 

98 1 8 
5821 
1811 

7789 

3755 
9710 

5653 
158;; 

75°4 

3412 

9309 

5'94 
1068 
6931 

2782 
8622 

4450 
0268 

6074 
1869 

7653 
34*7 
9189 
4940 

068 1 
641 1 
2130 
7838 
3535 

9222 

4899 
0565 
6220 
1865 

7499 
3113 

»737 
4340 

9933 

5516 
1089 
6652 
2204 

7747 

3J79 
8802 

43'5 
9817 
S3'o 

0793 
6267 
1730 
7184 
2628 

8063 

8903 
4309 
9705 


0.169  5092 


6014 
6003 
5990 

5978 
5966 

5955 
5943 

5';32 
5')i9 
5908 

5897 
5885 

5874 

I585' 

15840 
5828 
5818 
5806 
5795 

5784 
15774 

575« 
1574' 

5730 

57'9 
15708 

5697 
15687 

'5677 
5666 

5655 
5645 
5634 
5624 

5614 
5603 

5593 
5583 

5573 
5563 

15552 
5543 
5532 

5523 
5513 

5502 

5493 
5483 

5474 
5463 
5454 
5444 
5435 

5425 
5415 
5406 
5396 
5387 


1 

0.240 

0. 

.241 

.242 

•*43 

.244 

0.245 

0. 

.2+6 

. 

.247 

.248 

.249 

0.250 

0. 

.251 

.252 

•*5  3 

.254 

0.155 

0. 

.256 

•*57 

. 

.258 

.259 

0.260 

0. 

.261 

.262 

. 

.263 

.264 

0.265 

0. 

.266 

.267 

.268 

.269 

0.270 

0. 

.271 

• 

.272 

•*73 

, 

.274 

0.275 

0. 

.276 

• 

.277 

. 

.278 

.279 

0.280 

0. 

.281 

.282 

.283 

.284 

0.285 

0. 

.286 

.287 

.288 

.289 

0.290 

0. 

.291 

.292 

.293 

.294 

0.295 

0. 

.296 

.297 

.298 

.299 

0.300 

0. 

U>g  «3 


.169 

.170 
.170 

•17' 
171 

172 

172 

«73 

173 

'74 

•'74 

•'75 
•'75 
.176 
.176 

•'77 

178 

.178 

'79 
•'79 

180 
.180 
.181 
.181 
.182 

1S2 

184 
184 

'!« 
'85 
186 
186 

187 

187 
188 
18S 
189 
189 

190 
190 
191 
191 
192 

192 
'93 
93 
194 
194 

•'95 
.195 
.196 
.196 
•'97 

•'97 
.198 
.198 
'99 
•'99 


5092 
0470 
5838 
1197 

6547 

1887 
7218 
2540 

7853 
3156 

8451 

3736 
9013 
4280 
9538 

4788 
0029 
5261 
04S4 
5698 

0903 
6100 
1288 

6467 
1638 

6800 

'953 
7098 

2235 

7363 

2483 

7594 
2696 

779' 
2877 

7955 
3024 
8085 
3138 
^183 

3220 
8249 
3269 
8281 
3286 

8282 

3271 
8251 
3224 
8188  I 

3'45  ! 
8094; 

7968  : 
"94  i 

78"! 
2721  I 
7624  I 
2518  I 
7406 


200  2285 


Ditr. 


5378 
5368 

359 

350 
340 

331 

322 

3'3 

303 
295 

28  5 

277 
267 
258 
250 

241 
232 
223 

214 
205 

'97 
188 

'79 
'7' 
162 

'53 
'45 
137 
128 
120 

1 11 

102 
095 
086 

078 

069 
061 
°53 
045 

037 

029 
020 
012 
005 
4996 

(4989 

'4980 

4973 
4964 

'4957 

4949 
494' 
4933 
4926 

4917 
4910 

4903 
4894 
4888 
4879 


I 


logi« 


0.300  I  0.200 

301  '  .200 

,302  .201 

303  .201 

.304  .202 


505  0.202 

306  .203 

307  .203 

308  .204  1050 

309  .204 


1285 

7'57 
2021 
6878 

1727 

6569 
1403 
6230 


310 
311 
312 

3'3  , 

3'4| 

3'5| 
316  I 
3'7  , 
318, 

3 '9  1 


0.205 
.205 
.206 
.206 
.206 

0.207 

.207 
.208 
.208 
.209 

I 

320  I  0.209 

321  t  .210 

322  i   .210 

323  j   .211 

3-4  !  ^2" 


325 
326  I 

3*7  i 
328' 
329 

330  I 
33'  ! 

332  i 

333  I 
3  34  ! 

336 
337 
338 
339 

34° 
341 
342 
343 

344 


0.212 
.212 
.213 

•2'3 
.214 

0.214 
.214 
.215 
.215 
.216 

0.216 

.217 
.217 
.218 
.218 

0.219 
.219 

.220 
.220 
.220 


345  I  o-"' 

346  I  .221 

347  !  •2*2 

348  !  .222 

349  i  ^223 


35° 
35' 
352 
353 
354 

356 

357 
358 
359 
360 


0.223 
.224 
.224 
.225 
.225 

0.225 
.226 
.226 
.227 
.227 

0.228 


5862 
0667 

5464 
0254 

5037 
9813 

4581 
9342 
4096 
8843 

3582 

83.5 
3040 

7759 
2470 

7'74 
1871 
6562 
1245 
5921 
0591 

5253 
9909 

4558 

9210 

3835 

8464 

3085 

7700 

2308 
6910 

1505 
6093 
0675 

5250 
9818 

4380 
8935 
3483 

8025 
2561 

7090 
1613 
61  30 
0640 

5143 
9640 
4131 
8615 
3093 
7565 

2031 


Ditr. 


4872 
•  4864 

14857 

1  4849 
I  4842 

J4834 

;  4820 
'  4812 

J4805 

j  4797 

I  4790 

478J 

4776 

4768 

4761 
i  4754 
4747 
4739 
4733 

4725 
47 '9 
47" 
4704 
4697 

4691 

4683 
4676 
4670 
4662 

I  4656 

■  4649 
j  4642 

14635 
I  4629 

''  4621 

4615 
4608 
4602 

'  4595 
4588 

.4582 

4575 
4568 

j  4562 

''  4555 
4548 
4542 
4536 
4529 

4523 
45 '7 
45'° 
45°3 
4497 

4491 
4484  I 
4478 

4466 


027 


TABLE  XIII. 

For  finding  the  Ratio  of  llie  Swtor  to  tlu>  Triangle, 


logd" 


Din. 


0.560  0.228 
.361  .22X 
.362 

•363 
.364 


0.365 
.366 
.367 
.36S 
.369 

0.370 
•371 
•37» 
•373 
•374 

o-375 
•376 
•377 
.37S 

•379 

0.380 
.38. 
.382 

■383 
.384 

0.38s 
.386 
.387 
.388 
.389 

0.390 
.391 
.392 
•393 

^'•395 
V)6 

•397 
•398 
•399 

0.400 
.401 
.402 
.403 
.404 

0.405 
.406 
.407 
.408 
.409 

0.410 
.411 
.412 
.413 


.229 
.229 
.229 

0.230 
.230 
.231 
.231 
.232 

0.232 
•*3  3 
•*33 
.233 
.234 

0.234 
.235 
.235 
.236 
.236 

0.237 
.237 
.237 
.238 
.23S 

0.239 
.239 
.240 
.240 
.240 

0.241 
.241 
.242 
.242 
.243 

0.243 
.243 

•24+ 
.244 
.245 

0.24s 
.245 
.246 
.246 

.247 


2031 
6490 

0943 
5390 

9831 

A265 

5694 
3116 

753* 
1942 

6346 
0743 
5135 
9521 
3900 

8274 
2642 

7003 

'359 

5709 

0053 
439' 
X723 
3050 

7370 

1685 

5993 
0296 

4594 
8885 

3171 

7451 
1725 

5994 
0257 

45'4 
8766 

3012 
7252 
1487 

5716 
9940 

4158 
8371 
2578 


0.247  6779 

.248  0975 

.248  5166 

.248  9351 

•249  3531 

0.249  7705 

.250  1874 

.250  6038 

.251  0196 


4459 
4453 
i4447 
4441 
4434 

'4429 
4422 
'4416 
I4410 
,4404 

14397 
'43/2 
43^6 

143/ 9 
:4374 

14368 
4361 
4356 
4350 
4344 

4338 
4332 
4327 
4320 

43«5 

4308 

43°3 
4298 

4291 
4286 

4280 

4274 
4269 
4263 
4257 

4252 
4246 
4240 

4235 
4229 

4224 
4218 

4213 
4207 
4201 

c  4»96 


4191 

J4185 
4180 
i4»74 
I4169 

:4>64 
;4i58 
14153 

,4>47 


252  2^38  ;4;42 
•252  6775 


.414  I  .251  4349 

1.415  o.;;5i  8496 

.416   .252  20 

.417 

.418 

.419 

1.420 


.253  090 

•253  5°32 

0.253  9'53 


4'37 
6'4'3 


4126 
4121 


0.420 
.421 
.422 
•423 
•424 

425 
426 

427 
428 

429 

430 

43' 

43?. 

433 
434 

■t55 

43' 

43-' 

438 

439 

440 

441 

442 

443 

444 

445 
446 

447 
448 
449 

450 
45« 
452 

453 
454 

455 
456 

457 
458 

459 

460 
461 
462 
463 
464 

465 
466 

467 
468 
469 

470 

471 
472 
473 
474 

475 
476 

477 
478 

479 

480 


lo«,VJ 


0.253 

•254 
.254 

•255 
.255 

0.255 
.256 
.256 

•257 
•257 

0.258 
.258 

.,.58 

•259 
.259 

0.260 
.260 
.260 
.261 
.261 

0.262 
.262 
.262 
.263 
.263 

0.264 
.264 
.264 
.265 
.265 

0.266 
.266 
.266 

.267 
.267 

0.268 
.268 
.268 
.269 
.269 

0.269 

.270 
.270 
.271 
.271 


9' 5  3 

3269 

7  379 
1484 

5584 

9679 
3769 

7853 
1932 
6006 

0075 

4»39 
8198 
2252 
6300 

0344 
4382 
8415 

2444 
6467 

0486 

4499 

8507 

6509 

0503 
4492 

8475 

2454 
6428 

0397 
4362 
8321 
2276 
6226 

0171 
4111 

8046 
1977 
5903 

9824 

3741 
7652 

'559 
5462 


Diir. 


4116 
4110 

4«o5 
4100 

!4°95 

1 4  090 
4084 

14079 
i4074 
4069 

4064 
4059 

I4054 
14048 

4°44 

4038 

4033 
4029 
4023 
4019 

4013 
4008 
4004 
3998 

3994 
3989 
3983 
3979 
3974 
3969 

3965 
3959 
3955 
395° 
3945 

3940 
3935 
393' 
3926 

3921 

I 

39«7 
391 1 
3907 

3903 
3898 

3893 


0.271  9360 

1 3884 

3879 
3874 

3870 
3865 
3861 
3856 
3852 

3847 
3842 
3838 

3834 


.272  7141 
.273 1025 
•273  49041 
0.273  8778 1 

.274  2648 ! 
•274  6513 
•275  0374 
.275  4230 

0.275  8082 
.276  1929 
.276  5771 
.276  9609 


■277  3443  3^8^5 


0.277  7272 


628 


lug  «' 


- 
480 

0.277 

7272 

481 

.278 

1096 

482 

.278 

49 1 6 

483 

.278 

8732 

484 

.279 

2543 

485 

0.279 

6349 

486 

.2X0 

0151 

487 

.280 

394'» 

488 

.280 

7743 

489 

.281 

1532 

490 

0.281 

5316 

491 

.281 

9096 

492 

.282 

2872 

493 

.282 

6644 

494 

.283 

041 1 

495 

0.283 

4'73 

496 

.28^ 

7932 

497 

.284 

1686 

498 

.284 

5436 
9181 

499 

.284 

500 

0.285 

2923 

501 

.285 

6660 

502 

.286 

0392 

503 

.286 

4121 

5°4 

.286 

7845 

505 

0.287 

1565 

506 

.287 

5281 

5°7 

.287 

8992 

508 

.288 

2700 

509 

.288 

6403 

510 

0.289 

0102 

51' 

.289 

3797 

512 

.289  7487 

513 

.290 

"74 

4856 

514 

.290 

S'5 

0.290 

8535 

516 

.291 

2209 

5'Z 

.291 

5879 

518 

.291 

9545 

519 

•292 

3207 

520 

0.292 

6864 

051! 

521 

•293 

522 

•293 

4168 

523 

•293 

7813 

524 

•294 

'455 

525  '  0.294  5092 1 


526 


.294  8726 


527  1  -295  2355 

528  .295  5981 


529 


195  96c 


530  I  0.296  3220 

.296  6833 


53' 
532 
533 
534 

535 
536 

537 
538 
539 
540 


•297  0443 
•297  4049 
.297  7650 

0.298  1248 
.298  4842  3.594 
.298  8432 


359 


.299  201 


8^35 


.299  5600 
0.299  9*78 


3582 

3578 


1)1  ir. 


3824 

3820 
3816 
38,1 
3806 

3802 
3798 
3794 
378<; 
3784 

3780 
3776 
3772 

3767 
3762 

3759 

3754 
3750 

3745 
3742  I 

3737  I 
3732  i 
3729 

3724 
3720 

3716 
37"  , 

3708  ; 

3703  : 

3699  i 
3695  i 

3690  ! 
3687 
3682 
3679 

3674 
3670 
3666 
3662 
3657  j 

3654! 
365°  i 
3645 
,  3642  I 
I  3637 

3634  : 
3629 
3626  ! 
3621  i 
36,8  j 

3613  i 

3610 

3606 

3601 

3598 


ntff. 


3824 

3806  j 

3802 

,     379« 

3794 

378<; 

■     3734 

3  i  378° 
,  i  3776 
-i  3772 
\  3767 
376a 

3I 

2 

6 

16 

li 

'3 

30  I 
JJ] 
ZI 

45 

§sl 

9*  I 

'oo  ! 

r°3 


35 
09' 

79 
45 

•07  ', 

68  i 
13: 

•55  1 

192  1 
26 

55 

.81 
02 

20 
33 
43 
'49 


3759 
3754  ! 

3750  ' 
3745 
374*  ) 

3737 

373* 
3729 
3724 
3720 

1  37»6 

37'' 
3708 
3703 
3699 

3695 
3690 
3687 
3682 
3679  I 

3674' 
3670 
3666 
3662 

;  3657 

;  3654 

!    3650 
1    3645 
,    364*  I' 
I    3637   j 

3634  ' 

3621 1 
3618 

3613 
3610 
3606 
3601 


,50 
48 

}32i 
18  ': 
00] 

78 


i  359» 

3594 

359° 
3586 

3578 


TABLE  XIII. 

For  fmilint^  the  I{;itio  of  i\w  Sector  to  tlu-  TriiiiiKlc. 


1 

luK-.l 

0.540 

0.499  9'7'' 

54' 

.300  2752 

54» 

.300  6323 
.300  9890 

543 

544 

.301  3452 

0 

545 

0.301  7011 

546 

.302  0566 

547 

.302  4117 

54« 

,302  7664 

549 

.303  J208 

0 

550 

0.303  A748  : 
.303  828J. 
.304  1816 

551 

55* 

553 

•3°4  5344 

554 

.304  8869 

0 

555 

C.305  2390 

55b 

•305  5907 

557 

.305  9420 

55« 

.306  2930 

559 

.306  6436 

0 

560 

0.306  9938  , 

I  HIT. 


3574 
357' 
3  5<'7 
3562 

3559 

3555 
355' 
3547 
3544 
3540 

3536 

35  3* 
3518 

3515 
35*' 

35'7 

35'3 
3510 
3506 
3502 


i'>H.i^ 


0.560 
.^61 

.562 
.563 
.564 

0.565 
.566 
.567 
.568 
.569 

o.;70 
•57' 
■57» 
•573 
•574 

0.575 
.576 

•577 
.578 

•579 

0.580 


0.306 

.307 
.307 
.308 
.308 

0.308 
.309 
.309 
.309 
.310 

0.310 
.310 
•3" 
•3" 
,311 

0.312 
.312 
.312 
•3'3 
■3'3 

0.313  9215 


9938 
3437 
69  3  • 
0422 
3910 

7394 

0874 

4350 
7823 
1292 

4758 
8220 
1678 

5'33 
8584 

2031 

5475 
8915 

1351 
5785 


niff. 


3499 
3494 
349" 
3488 

3484 
3480 
3476 
3473 
3469 
3466 

3462 
3458 
345  5 
345' 
3447 

3444 
3440 
3437 
343  3 
3430 


0.5S0 
.581 
.582 

•583 
.584 

0.585 
.586 

■587 
.588 
.589 

0.590 

•59' 
.592 

•593 
•594 

0.595 
.596 

•597 
.598 

•599 

Q.600 


log  i« 


i    Piff. 


3'3 

3'4 
3'4 
3'4 

•'5 

3'5 
3'5 
3.6 
316 
316 

3'7 

3'7 
318 
318 
318 

3'9 
319 

3'9 

320 

320 


9115 

26a  I 
6064 

9483 

2898 

6310 
97 '9 

6525 
9913 
3318 
6709 
0096 
3480 
6861 

0238 
3612 
6983 
0350 
37'4 


320  7074 


3426 

34M 
34"  9 
34'5 
3412 

3409 
340^ 
3401 
3398 
3395 

3391 

3387 

3384 
338, 

3377 
3  374 

3367 

3364 
3360 


TABLE  XIV. 

For  finding  the  Kalio  of  the  Sector  to  the  Triangle. 


Elliiiac.   ! 

0.000 

0.000 

0000 

.001 

.000 

0001 

.002 

.000 

0002 

.003 

.000 

0005 

.004 

.000 

0009 

0.005 

0.000 

0014 

.006 

.000 

0021 

.007 

.000 

0028 

.008 

.000 

0037 

.009 

.000 

0047 

O.OIO 

0.000 

0058 

.oil 

.000 

0070 

.012 

.000 

0083 

.013 

.000 

0097 

.014 

.000 

0113 

0.015 

0.000 

0130 

.016 

.000 

oia8 
0167 

.017 

.000 

.018 

.000 

0187 

.019 

.000 

0209 

0.020 

0.000 

0231 

.02  1 

.000 

0255 

.022 

.000 

0280 

.023 

.000 

0306 

.024 

.000 

0334 

0.025 

0.000 

0362 

.026 

.000 

0392 

.027 

.000 

0423 

.028 

.000 

0455 

.029 

.000 

0489 

3.030 

0.000 

0523 

Diff.         Iljporboln. 


I 
I 

3 
4 
S 

7 
7 
9 

10 
II 

12 
'3 
14 
16 

'7 
18 

19 

20 
22 
22 

*4 
*5 
26 
28 
28 

30 
3' 
3» 
34 
34 


Diff. 


0.000  0000 
.000  0001 
.000  0002  j 
.000  0005 
.000  0009 

0.000  0014  ; 
.000  0020  i 
.000  0028  ' 
.000  0036  I 
.000  0046  i 

0.000  0057  ; 
.000  0069  I 

.000  0082  ' 

.000  0096  1 
.000  01  II 


0.000 

.000 
.000 

.000 
.000 

0.000 
.000 

.000 

.000 

.000 


0127 

0145 

0164 

0183 

0204 

0226 
0249 

0273  : 

0298  i 
0325  ! 

0.000  0352 
.000  0381  I 
.000  0410  ; 
.000  0441  '' 
.000  0473  ! 

0.000  0506  ' 


10 
II 

12 
'3 
'4 
'5 
16 

18 
19 
'9 
21 

22 

23 
S4 
15 
*7 
27 

29 
29 
3' 
3» 
33 


0.030 
.031 
.032 
.033 
.034 

0.035 
.036 
.037 
.038 
.039 

0.040 
.041 
.042 
.043 
.044 

0.045 
.046 
.047 
.048 
.049 

0.050 
.051 
.052 
.053 
.054 

0.055 
.056 
.057 
.058 
.059 

0.060 


t 

s 


EIII118P. 


0.000  0523 
.000  0559 
.000  0596 
.000  0634 
.000  0674 


0714 
0756 
0799 
0844 
0889 


0.000 
.000 
.000 
.000 
.000 

0.000 
.000 
.000 
.000 
.000 

0.000 

.000 
.000 
.000 
.000 


0936 
098.1 
1033 
1084 
"35 

1188 
1242 
1298 

'354 
1412 


0.000  1471 

.000  1532 

.000  1593 

.000  1656 

.000  1720 

0.000  1785 

.000  1852 

.000  1920 

.000  1989 

.000  2060 

0.000  21 31 


Difr. 


36 

38 
40 
40 

4a 
43 
45 
45 
47 

48 
49 

5' 
5' 

53 

^t 
^f, 
^t 
58 

59 

61 
61 

64 
65 

67 
68 
69 

71 
7' 


Iljiiorliola. 


0.000 
.000 
.000 

.000 
.000 

0.000 
.000 
.000 
.000 
.000 

0.000 
.000 
.000 
.000 
.000 

0.000 
.000 
.000 
.000 
.000 

0.000 
.000 
.000 
.000 
.000 

0.000 
.000 
.000 
.000 
.000 

0.000 


0506 
°539 

0575 
061 1 

0648 

0686 
0726 
0766 
0807 
0850 

0894 
0938 
0984 
1031 
1079 

1128 
1178  , 
1229 
1281  I 

'334  i 

1389! 

'444! 
1500 

'558! 
1616  I 

1675 
1736 
1798 
i860 
1924 

1988 


DifT. 


36 

38 

40 
40 

4' 
43 

44 

tt 

ii 

49 
50 

5' 

5» 

S3 
55 

'! 

Is 
59 

61 

62 
62 

64 
64 


62U 


TABLE  XIV. 

For  (ii)(Iing  Uie  ]\;itii)  of  the  Sei'tor  ici  tlie  Triangle. 


^ 


KlliiMi!.  Diff.         "I,vi«'iI)oItt.    I     Din. 


0.060 
.061 
.062 
.063 
.064 

0.065 
.066 
.067 
.068 
.069 

0.070 
.071 
.072 
.073 
.074 

0.075 
.076 
.077 
.078 
.079 

0.080 
.081 
.0S2 
.0S3 
.084 

0.085 
.086 
.087 
.088 
.089 

0.090 
.091 
.092 

•093 
.094 


o.cg5  I  0.000 
.096  I    .coo 


0.000 
.oco 
.000 
.000 

.000 

0.000 
.000 
.000 
.coo 

.000 

0.000 

.000 

.000 

I     .000 

i     .000 

I  0.000 
.oco 
.000 
.000 
.coo 

0.000 
.000 
.000 
.000 

.000 
0.000 

.000 
.coo 
.000 

.000 

0.000 
.coo 
.000 

.000 

.coo 


.098 
.099 


000 

CJO 

oco 


o.ioo  I  0.000 
.101  :  .ooo 
.102  [  .000 
.103  I     .coc 

.104 


.000 


0.105 
.106 

.107 
.108 
.109 

o.iio 
.II I 

.112 

•"3 
114 

o.ii; 

.116 

.117 
.118 

,H9 


0.000 
.000 
.000 
.000 
.000 

coco 

.000 

.000 
.oco 
.oco 

0.000 

.000 

.000 
.000 
.000 


2131 

2204 

2278 

2354 
2431 

2509 
2588 

2669 

-751 

2834 

2918 

3004 
3091 
3180 
3269 

3360 
3453 
3546 
3^*4' 
3738 

3«35 
3934 
4034 
4136 

4139 

4343 
4448 

4555 
4663 

4773 

4884 

4996 

5109 

S^H  , 

5341 

5458 

5577 
5697 
5819 
5941 

6066 
6192 

<^3'9 
6448 

6578 

6709 
6842 
6976 
7111 

7248 

7386 

7526 
7667 
7809 

7953 
8098 
8245 

8393 
8542 
8693 


73 

7-, 
76 

77 
78 

79 

81 
82 

84 

86 

87 

l9 
89 

91 

93 
93 
95 

97 
97 

99 

CO 

02 
°3 
«4  I 

05  I 

07  I 

08  I 
10 

I  I 


O.I  20  I  O.OOC  8845 


12 

13 
15 

17 

J7 

'9 

20 

23 
24 

26 

2  7 

19 

30 

31 

33 
34 
35 
37 
38 

40 
41 
42 
44 
45 

47 
48 

49 
S« 

51 


0.000 
.oco 
.000 
.000 
.000 

0.000 
.coo 
.oco 
.000 
.000 

0.000 

.000 
.000 
.000 
.coo 

0.000 

.000 

.000 
.000 

.000 
0.000 

.000 

.coo 
.000 
.000 

c.coo 

.000 

.000 

.000 
.000 

0.000 
,000 

.000 

.000 
.000 

0.000 
.000 
.000 
.000 

.000 

0.000 
.000 

.000 

.000 

.000 

0.000 
.000 
.000 

.000 

.000 

0.000 

.000 
.000 
.000 
.000 

0.000 

.000 

•coo 
.000 
.000 

0.0  DO 


1988 1 
2054 ; 

2I2I 

2189  : 

2257 1 

2327  j 
2398 
2470 1 
2543 

2617 
2691 

2767 

2844 
2922 

3001 
:,o8i 

3162 

3244 
3327 

34" 

3496 1 
3582  j 
3669  ! 

3757  : 

3846; 

393*' 

4027  j 
4119 

4212  - 
4306  j 

4401 
449(1  i 

4n93 
4691 

4790 

4890  ' 

4991 

5092 

5  "95 
5299  , 

5403  ' 

55'^9 

5616 

5723  I 
5852  I 

594» 
6052 
6163 
6275 
6389 

0503  ■ 
6618 

6734 
6851 
6969 

7088  ■ 
7208  ' 
7  3 '^9 
745  > 
7574 

7<''9^' 


66 

67 
68 
68 

70 

71 

72 
73 
74 
74 
"6 

77 
78 

79 

80 

81 

82 

83 
84 

85 

86 
87 
88 
89 
90 

91 
92 

93 
94 

9? 

95 
97 
98 

99 

100 

lOI 

10 1 

103 
104 
104 

106 
107 
107 
109 
109 

III 
II I 
112 
114 
114 

115 

116 

H7 

118 
119 

120 
1 2 1 

122 

123 
15.4 


Klli|.sc.       I     Diff.         lI.v|)(rbolu.         Diff. 


20  j  0.000 


2  I 


24 

25 

26  I 

27  I 

28  I 


29  ] 


.000 

i     .000 

I 

I     .000 

I     .oco 
}  0.000 

.000 
i       .000 

.00 
.00 


30 
31 

32 

53 
.34 


I 


0.00 
.00 
.00 

.00 

.00 


35  I  0.00 

36  I    .00 

37  '    .00 

38  I    .00 

39  i     -oo 

40  I  0.00 

41  !    .00 


.00 
.00 
.00 


42 
43 
44 

45  o.co 

46  .00 

47  I     .00 

48  ;     .00 

49  :      .00 

50  o.co 

51  :  .00 

52  \  .00 

53  I  -O" 

54  ,  -co 

0.00 
.00 
.00 
.00 
.00 


5'^  : 
56 

58! 

59 

60  i  0.00 

61 

62 

63 
64 

65 
66 
67 
68 
69 

7= 
71 

7'-' 
73 
74 

75 
76 
77 
78 
79 
1.^0 


.00 

.00 

.00 
0.00 

.00 
.00 

.00 

.00 
0.00 

.00 
.00 

.oc 

.00 

o.co 

.00 
.00 
.002 
.002 

0.002 


8845 

8999 

9»54 
93 1 1 

9469 

9628 
9789 
9951 

CI  15 

0280 

0447 

061  5 

0784 

0955 
1128 

1301 

'477 
1654 
1832 
2012 

2193 

2376 
2560  i 
2745 
2933  I 
3121  I 
33««  I 
3';o3  i 

3696 1 
389'  j 

4087  j 
4285 

4484 
4684 

4886  I 

5090  j 
5295 1 

5502  - 

5710 
5920 ; 

6131 
6344  i 
6559 

'>775 
6992 

721  (   ; 

7432 
7654  : 
-S78 

8103  ; 

8330  I 
8558  I 
8788  I 
9020  I 
9253  { 

9487 
9724 
9961 

02Cl 
0442 

'^f'85 


»54 

155 
'57 
158 

'59 

161 
162 

164 
165 
167 

168 
169 
171 

173 
'73 
176 

'77 
178 
180 
181 

•83 
184 
185 
188 
188 

190 
192 

'93 

195 
196 

198 
199 

200 

202 
20,). 

205 

207 

io8 
210 
211 

i'3 

2'5 

216 
217 
Z19 

221 

22X 
224 
225 

227 

228 
230 
232 

"  iS 

234 
237 

'37 
240 
241 
243 


7698 

7822 

794!< 

8074, 
8202  I 

8330' 

8459  ; 
8590  ; 
8721  i 
8853  j 
8986  I 
9120  j 
9255  I 
9390 
9527  j 

9665  i 
9803  i 

9943  ] 
0083  ! 

0224  I 

0366  I 
0509  I 
0653  [ 

0798  I 
0944  j 

1 09 1  i 
1238  i 

1387' 
1536  ' 
1686 

,838  j 
1990 
2'43 : 

2296  I 

2451  I 

2607  I 
2763  ! 

292T  : 

3C79  ! 
3238 , 

3398  j 

3<i59  I 
3721 

3883 

4047 

421 1 

4377 
4543  I 
4710  I 
4878 

5047 
5216  I 

5387 
5558 

5730  I 

5903 ! 
6077 ' 
6252 ' 
6428 
6604 

o.coi  6-82 


j  0.000 
i  .000 
.000 
I  .000 
I  .000 

I  0.000 

I  .000 

I  .coo 

.000 

.000 

0.000 

.000 

.000 

.000 
.000 

0.000 
.000 

.000 

.001 

.OOI 

O.OOI 
.001 

.001 

.001 

.001 

O.OOI 
.001 
.001 

.001 
.001 

O.OOI 
.001 
.001 
.001 
.001 

O.OOI 
.001 
.001 
.001 
.001 

I  O.OOI 
j   .001 


.001 

.001 
.001 

I  0.00: 

I    OOI 

I  .001 
I  .001 

I   .001 
I  O.OOI 

I  .001 

.001 

.001 

.001 

O.OOI 

001 

I   .001 

I   .001 

I  .001 


24 
26 
26 
28 
28 

29 

3' 
3' 

32 

33 

34 

35 
35 
37 
38 

38 

40 
40 

41 

42 

43 
44 
45 
46 

47 

47 
49 
49 
5° 
52 

52 
53 
53 
55 
56 

'^8 

58 

59 
60 

61 

62 
62 
64 
64 

66 
66 

67 
68 
69 

69 

71  i 

7' 

72 
73 

74 
75 

76 
-i 


C'JO 


pcrbola. 

30    7698   1 
30    7S22 
30    7948   I 
00    8074  j 

00   8202  j 

i 

00  8330 
00  8459  ■ 
00  8:590  ; 

:00      8721     I 

100  8853  I 

iOO  8986  1 
)oo  9120  ! 
joo  9255  i 

DOO  9390  ' 

:oo  9527  ! 

000  9665  I 
000  9803  j 

000  9943  j 
coi  0083  I 

00 1  0224 

001  0366 
001  0509 

OOT  0653  I 

,oot  0798  I 

.001  0944  I 

.00!  1091  ' 
,COI   1238  '; 

,001  1387 
.001  1536  ' 
.001  i()86  ' 

.001  1838  I 
.001  1990 
.001  2143 
.001  2296 
01  2451 

001  2607 
001  2763  i 
001  2921 

3079 ! 
3^-38 , 

001    3395! 

3S 


DifT. 


124 
126 
126 
128 
128 


.001 
.001 


.001 

■  OOt 
.001 


3721 

3«!<3 


.coi  4047 

1.00".  421 1  j 
.joi  4377  '■■ 
.001  4S43  I 
.001  4710  1 
.001  4878  ! 

1. 001  504'; 
.001  5216  1 
.001  5387  I 

.001  5558 ; 
31  5730 1 

1. 001  5901  ! 
joi  O077  \ 
.001  62^2  I 
.001  6418  j 
.001  6604  i 


129 

131 
13- 
133 

134 
135 

«3i 

137 
138 

138 

140 
140 
141 
142 

143 
144 
U5 
146 

147 

147 
149 
149 
150 
152 

152 

153 
I  153 

«55  ; 

.56 ' 
156 ' 

159 
160 

161 
162 
162 
164 
164 

16(1 

i68 
169 

169 
171 
171 
I"" 
173 

'74 
«7> 
i-f' 
176 

I 


TABLE  XIV. 

For  finding  tlic  Iviitio  of  tla-  Sector  to  the  Triangle. 


>.ooi   678a 


i 

Kllipse. 

Diff. 

Ilyperbolii. 

Diir. 

178 
179 
180 

0.180 

0.002 

0685 

O.OOI 

6782 

.iSi 

.182 

'  .1X3 

,002 
.002 
.002 

0929 

i>75 
1422 

244 
246 
247 

.001 
.001 
.001 

(i960 

7139 
7319 

.184 

.002 

1671 

249 

.001 

7500 

iSl 

'  0.18; 

0.002 

1922 

0.001 

7681 

.186 
.187 

.002 
.002 

2174 
2428 

252 
»54 

.001 
,001 

7864 

8047 

1^3 
'83 

;   .188 
.1S9 

.002 
.002 

2(183 
2941 

255 
258 
258 

.001 

.001 

Si  3 1 
8416 

iS4 
185 
:86 

,  0.190 

0.002 

3 '99 

261 
262 
263 
266 

O.OOI 

8602 

187 
187 
189 
1S9 
190 

'   .191 
.192 

.002 
.002 

3460 

3722 

.001 
,001 

8789 
8976 

.193 

.002 

3985 

.001 

9165 

!  ''94 

.002 

4251 

267 

.001 

9354 

0.195 

0.002 

451S 

268 
270 

O.OOI 

9544 

191 
191 

.196 
.197 

.002 
.002 

4786 
5056 

.COI 

.001 

9735 
9926 

.198 

.002 

5328 

272 

.002 

0119 
0312 

'93 

.199 

.002 

5602 

274 
275 

.002 

193 
J95 

0.200 
.201 

0,002 
.002 

5877 
6154 

177 

0.002 
.002 

0507 
0702 

«9; 

.202 
.203 
.204 

.002 
.002 
.002 

6433 
6713 
6995 

279 
280 
282 
283 

.002 

.002 
.002 

0897 
1094 
1292 

'95 
'97 
,98 
19S 

.  0.205 
.206 
.207 
.208 

0.002 
.002 
.002 
.002 

7278 

7564 
7S5, 
8139 

286 

287 
288 

C.O02 
.002 
.002 
.002 

1889 
2090 

199 

?oo 
201 

.209 

.002 

8429 

290 
2(;j 

.002 

2:91 

201 

203 

0.210 
.211 
.212 

0.002 
.002 
.002 

8722 
9...T5 
9311 

207 
296 

0.002 
.002 
.002 

2494 
2(197 
2901 

203 
204 

.213 

.002 

9608 
9907 

297 

.002 

3106 
331' 

205 

.214 

.002 

299 

300 

.002 

205 

207 

0.215 
■  .216 

I  -217 
.218 
.219 

1^.003 
.003 
.003 
.003 
.003 

0207 
0509 
OS;  t 
1 1 19 
1427 

302 
30s 

305 
308 
3C9 

0.002 
.002 
.002 
.002 
.002 

3518 

3725 

3931 
4142 

435i 

207 
207 
210 

2IO 
210 

0.220 
.221 
.222 

0.003 
.003 
.003 

1736 

2047 

^359 

311 
312 

316 
318 

0.002 
.002 

,002 

4774 
4986 

211 

212 

113 
*'5 

1  ••■^^3 
.224 

.003 
.003 

2674 
2990 

.002 
.002 

5«99 
54«i 

0.2?,:; 
.226 
.227 

0.003 
.003 
.003 

3308 
3(127 

3949 

3-0 

0.001 
.CCi2 
.00?. 

5627 
5842 
(1058 

216 

.228 
.229 

.003 
,003 

4272 
4597 

3^3 
3*5 

327 

.002 
.002 

6275 
6493 

217 
21s 
218 

0.230 
.231 

.232 
•■233 

0.003 
.003 
.003 

.00  ? 

4924 
5252 
5582 
59'4 

328 
330 
332 

0.002 
.002 
,002 

.002 

6711 
6931 
71  ;i 

7371 

220 
220 
220 

.234 

.003 

6248 

334 
336 

.002 

7593 

2  22 
223 

0'23.5 
.236 
.237 
.238 
•'•39 

0.003 

.003 
.003 
.003 

.C03 

6584 
6921 

7260 
7601 
7944 

337 
339 
34: 
343 
345 

0.002 
.002 
.002 
.002 
.002 

7816 
8039 
X263 
8487 
87,3 

223 
224 

224 
2  2fi 
22(1 

0.240 

0.003 

8289 

0.002 

8939 

c 


Ellipse. 


'I 


0,240  ]  0.003  8289 

.241  j     .003  8635 

.242   ;      .003  {',983 

.243   '      .003  9333 

.244  I      .003  9(185 


C.245 
.24(1 

•247 
.248 
,249 

0.250 


•15'  : 
.252 

•253 
•154 

0.155  I 
.256  1 

■257    ': 
.258    i 

•259    j 

0.260  I 

.261  i 

.262  ; 

.263  I 

.264  I 

0.265  ' 

.266  I 

.267  ' 

.268  j 

.269  I 

0.270 
.271 
•272 
•273 

•274 : 
0.275 
.276 
.277 
.2-8 
.279 

0.280 
.281 
.282 
.283 
.284 

0.285 
.286 
.287 
.288 
.289 

0.290 
.291 
.292 
.193 
.294 

C'-295 
.296 
.297 
.298 

•299 


0.004 
.004 
.004 
.004 
.004 

0.004 

.004 
.004 
.004 
.004 

0.004 
.004 
.004 
.004 
.004 

0.004 
.004 
.004 
.004 
.004 

0.004 
.004 

.0C4 
.004 
.004 

c.004 
.004 
.005 
.005 
.005 

0.005 
.005 
.005 
.005 
.005 

0.005 
.005 
.005 
.005 
.005 

0.005 
.005 
.005 
.005 
.005 

0.005 
.005 
.005 
.005 
.005 

0.006 
.006 
.006 
.006 
.006 


Diff.     1     IIjpi.-rlMilii. 


Diff. 


0039 
0394 
0752 
11 11  I 

'472  j 

'835  I 

2199 

2566  i 

2934 

33°5 

3677 

4051 
4427 
4804 
5184 

5566 
5949 
'1334  i 
6711 
7111 

7502  I 
7894  1 
S289  ; 
8686  I 
9085  I 

9485  I 
9888  I 
0292  I 

0699 ; 

1107  j 

'5'7  I 
1930 

2344 
2760 
3178 

359S 

4020 

4444 
4870 
5298 

5728 

6160' 

65941 

7010, 

7468 ! 

7908  i 
8350 

8795 
9241  ' 
9(189 

0139: 
o5<)i : 
1045 
1 502 

i960  : 


C.3OO       D.Ocd     2421 


346 
348 

350 
352 

354 

355 
35'^ 
359 
361 
363 

364 
367 
368 

37' 
372 

374 
376 

377 
380 
382 

387 
390 
391 

392 
395 
397 
399 
4c  o 

403 
404 
407 
408 
410 

4'3 
414 
416 
418 
420 

422 
424 
J  26 
428 
430 

432 
434 
436 

43S 
440 

442 
445 
44'' 
448 
450 

452 
454 
457 
4  =  8 

461 


0.002 

8939 

.002 
.002 

9166 
9394 

217 
228 

.002 
.002 

9(123 
9852 

229 
229 
231 

0.003 

0083 

.003 

03 '4 

231 

233 

.003 

054^ 

.003 

077S 

.003 

101 1 

233 

234 

0.003 
.003 
.003 

1245 
1480 
1716 

235 
236 

.003 

1952 

.003 

2189 

237 
238 

0.003 

2427 

.003 

1666 

239 

.003 

2905 

239 

.003 

3146 

241 

.003 

3387 

241 

241 

0.003 

3628 

.003 

3^*71 

243 

■>°3 
.003 

4114 

435« 

2-^3 
2^4 

.003 

4603 

-4l 
245 

0.003 
.003 

4848 
5094 

246 

24: 
248 

.003 
.003 

534^ 

55^9 

.003 

5838 

249 
249 

0.C03 

6087 

.003 

f'337 

250 

.003 

(1587 

.003 
.003 

6839 
7091 

252 
252 
253 

0.003 

7  344 

2'>J 

.003 

759'' 

.003 

7852 

2s4 

.003 
.003 

8107 
8363 

256 

257 

0.003 

.003 

8620 

8877 

=^V 

.003 

9135 

259 

260 
260 

1  -003 
.003 

9394 
9^' 54 

!  0.003 

9914 

-61 

1  .004 

0175 

26-' 

'  .C04 

0437 

.63 

263 
264 

.004 
.004 

0700 
0963 

0.004 
.004 

1227 
H91 

264 

266 

.004 

1757 

266 

.004 

2023 

267 

.004 

2290 

267 

c.004 
.004 
.004 

2826 
3095 

269 

169 

269 

271 
271 

.004 

3  3<'4 

.004 

3<'35 

0.004  39°^ 


(i.'il 


I  : 


w 


TABLE  XV. 

For  Elliptic  Orbits  of  j;reat  eccentricity. 


TABLE  XVI. 

For  Hyperbolic  Orbits. 


«  or« 

li.g«oi>rlon/V 

Diff. 

log  N 

Diff. 

7 
21 

36 

49 
64 

t  i.r  fi 

IdgfloW'OK^V 

Diff. 

9 
II 
II 
li 

log  N 



Diff. 

436     1 

450 
464 

47'; 

493 

o 

0 

1 

2 
3 

o.ooo  oooo 
.ooo  oooo 
.ooo  coco 
.ooo  oooo 

0 
0 

0 

0 

0.000  oooo 
.000  0007 
.000  0028 

.COG    0064 

0 

30 
31 
32 
33 

0,000   0066 
.000  0075 
.000   0086 
.000    0097 

0.000  6400 
.000   6836 
.ooo  7286 
.000  775c 

4 

.ooo  oooo 

0 

.000  01 1 3 

34 

.000   0109 

n 

.000  8229 

5 

o.ooo  ocoo 

0 

0.000  0177 

78 

92 

107 

120 

»3S 

35 

O.OOO    0122 

\l 

0.000  8722 

508 

5^3 
537     ' 

567 

6 

.ooo  oooo 

.000  0255 

30 

.000  0137 

.000  9230 

7 

.000  oooo 

0 

.coo  0347 

37 

.000    O153 

t8 

.000  9753 

8 

1     9 

1 

.000  oooo 
.ooo  ooo  I 

I 

0 

.000  0454 
.000  0574 

38 
39 

.000  0171 

.000    0190 

«9 

20 

.001   0290 
.001   0842 

I    10 

1    11 

18 

o.ooo  ooo  I 
.000  ooo  I 

.ooo    0002 

0 

I 

0 

0.000  0709 
.000  0858 
.000  1 02 1 

149 
163 
178 
191 
206 

40 
41 
42 

0.000  02 10 
.000  0232 
.000  0255 

22 

0.00 1    1409 
.001    1990 
.001   2586 

581    ! 
596    ■ 
611    i 

13 

.000    0OO2 

.000  1 199 

43 

.000  0281 

27 
29 

.001    3197 

626    1 

14 

,000  0003 

1 

.000  J  390 

44 

.000  0308 

.001   3823 

640    1 

1   »•"' 

0.000  0004 

I 

0.000  1596 

220 

45 

0.000  0337 

3' 

33 

36 
38 

40 

0.00 1   4463 

6.55     i 
670 
685     , 
700 
71S 

i    16 
1    17 
1    18 

.000  0005 
.000  0007 
.000  0009 

2 
2 
2 

.000  1816 
.000  2051 
.oco  2299 

235 
248 
263 

277 

4a 

47 
48 

.000  0368 
.000  04CI 
.000  0437 

.001   5118 
.001   5788 
.001   6473 

!   10 

.000    001  I 

2 

.000  2562 

49 

.000  0475 

.001  7173 

20 

!   21 

0.000  001  3 
.000  0016 

3 
3 
4 
4 

s 

0.000  2839 
.000  31  31 

292 

50 
51 

0.000  0515 
,000  0558 

43 

48 

SI 

0.001  7888 
.001  8618 

730 

744 
760 

775 
790 

1   22 

.000  0019 

.000  3437 

320 
334 
349 

52 

.000  0604 

.001   9362 

23 
24 

.000  0023 
.000  0027 

.000   3757 
.000  4091 

53 
54 

.000  0652 
.000  0703 

.002    0122 
.002    0897 

25 

0.000  0032 

5 
6 

0.000  4440 

363 

;02 

55 

0.000  0757 

60 

0.002     1687 

806 

2G 

.000  0037 

.000  4803 

50 

.000  0815 

.002    2493 

820 

27 

.oco  0043 

7 

.000  51 81 

57 

.000  0875 

6,1 

.002    3313 

836 
851 
866 

;   18 

.000  C050 

7 
9 

.000  5573 

407 
420 

58 

.000   0939 

68 

.002    4149 

2» 

.00c  0057 

.000  5980 

i»« 

.000   1007 

71 

.002    5000 

1   30 

0.000  0066 

0.000  6400 

00 

0.000   1078 

0.0C2  5?66 

' 

morn 

log  Q  or  log  Q' 

log  I.  Diff. 

loglmlfll.Diff. 

m  or  n 
0.10 

log  Q  or  log  C 

log  I.  Diff. 

loglmlfll.Diff. 

0.00 

0.000   OOCO 

2.1 149,, 

9.998   7021 

3.41256,, 

2.1046„ 

.01 

9.999  9870 

i.4>597.. 

2,1146,, 

.11 

.998   4308 

3.45326,. 

2.1025,.    ^ 

.02 

•999  9479 

2.71675,, 

2.1 142,, 

.12 

.998    1342 

3.49028,, 

2.1003,, 

•03 

.999  8828 

2.89259,, 

2.  II 37,, 

•13 

•997  8»23 

3.52423.1 

2.o978„ 

.04 

•999  79«7 

3-oi74«n 

2.II30n 

.14 

.997  4654 

3-55547" 

2.0952, 

0.05 

9,999  6746 

3.ii4«i.i 

2.1121,, 

0.15 

9.997  0936 

3-58453" 

2.0923^ 

.06 

.999  5316 

3.19290,, 

2.11I0„ 

.16 

.996  6971 

3.61 154" 

2.0892™ 

.07 

.999  3628 

3.25940,, 

2.IC97,, 

•17 

.996   2760 

3.63679,, 
3.66048,, 

2.0860,. 

.08 

.999   1082 

3.31687,, 

2.1082,, 

.18 

•995  8305 

2.0826, 

.09 

.998  9479 

3-3674S" 

2.1065,, 

.19 

•995   3608 

3.68276,, 

2.0790, 

O.IO 

9.998  7021 

3.41256,. 

2.1046,, 

0.20 

9.994  8671 

3^70378« 

2.0752,  J 

0.0035 
.0036 

.0037 

.0038 
.0039 

0.0040 
.0041 
.0042 

.0043 
.0044 

0,0045 
.0046 

.0047 

.0048 

.0049 

0.0050 
.0051 
.0052 

.0053 
.0054 

0.0055 
.0056 

.0057 
.0058 

.0  -0 
.cob  . 


032 


TABLE  XVII. 

For  special  Perturbations. 


9.  9',  '{' 


o.oooo 

.0001 
.0001 

.0003 
.0004 

0.0005 

.ocoh 
.0007 
.0008 
.0009 

O.OOIO 
.001  I 
.0012 
.0013 
.0014 

0.0015 
.0016 
.0017 
.0018 
.0019 

0.0020 
.0021 
.0022 
.0023 
.0024 

0.0025 
.0026 
.0027 
.0028 
.0029 

0.0030 
.0031 
.0032 
.0033 
.0034 

0.0035 
.0036 
.0037 
.0038 
.0031} 

0.0040 
.0041 
.0042 
.0043 
.0044 

0.0045 
.0046 
.0047 
.0048 
.0049 

0.0050 
.0051 
.0052 
■0053 
.0054 

0.0055 
.0056 
.0057 
.0758 

.0   -o 

.00b  ' 


For  positive  values  of  tlio  Argument. 


log/ 


BiiT.        log/',  log/"      DilT. 


0.477 

•477 
.476 
.476 
476 


1213 

0127 
9042 

7957 
6872 

5787 

4''02 

3618 

*534 
1450 

0367 
9284 
8201 
7118 
6035 


476 
476 
476 
476 
476 

476 
475 
475 
475 
475 

475  4953 
475  3^7' 
475  2789 
475  1707 
475  0626 

474  9545 
474  8464 
474  7383 
474  6303 
474  5223 

474  4143 
474  3063 
474  "983 
474  0904 
473  9815 

473  8746 
473  7667 
473  6589 
473  55" 
473  4433 

473  3355 
473  2178 
473  I20I 

473  o«24 
472  9047 

472  7970 
472  6894 
472  5818 
472  4742 
472  3666 


47^ 
472 

472 
471 
471 

47' 

471 
471 
471 
47 « 

471 

47 « 
470 
470 
470 
470 


2591 
1516 
0441 
9366 
8292 

7218 
6144 
5070 
3996 
2923 

1850 

0777 

97-^4 
8632 
7500 
(14S8 


086 

085 
0S5 
085 
085 

085 
084 
084 
084 
083 

083 
083 
083 
083 
082 

082 
082 
082 
08 1 
081 

081 

081 
080 
080 
080 

080 
080 
079 
079 
079 

079 

078 
078 
078 
078 

077 
077 
077 
077 
077 

076 
076 
076 
076 
075 

075 

075 
075 
074 
074 
074 
074 
074 
073 
073 

073 

073 
072 
072 
072 


0.301  0300 
.300  9431 
.300  8563 
.300  7695 
.300  6827 

5959 
5092 
4224 

3357 
2490 

1623 

0756 
9889 
9023 

8157 


0.300 
.300 
.300 
.300 
.300 

0.300 
.300 
.299 
.299 
.299 

0.299 
.299 

.299 

.299 
.299 

0.299 
.299 
.299 
.299 
.298 

0.298 
.298 
.298 
.298 
.298 

0.298 
.298 
.298 
.29S 
.298 

0.298 
.297 
.297 
,297 
.297 

0.297 
.297 
.297 
.297 
.297 

0.297 
.297 
.296 
.296 
.296 

0.296 
.296 
.296 
.296 
.296 

0.296 
.296 
.296 
.296 

•295 
.295 


7291 
6425 

5559 
4693 

3828 
2963 

20q8 

»233 

0368 
9504 

8639 

7775 
69 1 1 

6047 
5184 

4320 
3457 
2594 
1731 
0868 

0005 

9143 

8280 
7418 
6556 

5695 
4833 
3972 
3H0 
2249 

1388 
0528 
9667 
8807 
7946 

7086 
6226 
5367 

4507 
3648 

2788 
1929 

1070 
0212 

935  3 
8495 


:  869 
868 
868 
868 
868 

867 
868 
S67 
867 
867 

1867 
867 

,  866 
866 

I  866 

'  866 
!  866 
I  866 
i865 

;  865 

1 865 
865 
\  865 
1S64 
1 865 

'  864 
'  864 

864 
'  863 

864 

863 
863 

1 863 
863 

:  863 

■  862 
863 
862 
862 
861 

862 

86 1 

862 
861 
861 

860 
861 
860 
861 
860 

860 
859 
860 
859 
860 

859 
859 
858 
859 
858 


For  negative  values  of  tlie  Argument. 


log/ 


0-477 
•477 
■477 
•477 
•477 

0.477 

•477 

•477 

•+77 
■478 

0.478 
.478 
.478 
.478 
.478 

0.478 
.478 
.478 
•479 
•479 

0.479 
•479 
•479 
•479 
•479 

0.479 

•479 
.4S0 
.480 
.480 

0.480 
.480 
.480 
.480 
.480 

0.480 
.481 
.481 
.481 
.481 

0.481 
.481 
.481 
.431 
.481 

0.482 
.482 
.482 
.482 
.482 

.482 
.4S2 
.482 
.483 

0.483 
.483 
.483 
.483 
.483 
•483 


1213 
2299 
3385 
447 « 
5558 
6645 
7732 
8819 
9906 
0994 

2082 
3170 
4*59 
5348 
6437 
7526 
8615 
97"5 
0795 
1885 

2975 
4065 
5156 
6247 
7338 

8430 
9522 
0614 
1706 
2798 

'3891 
4984 
6077 
7170 
8264 

9358 
0452 

1547 
2641 

3736 

4831 
5926 
7022 
8118 
9214 

0310 
1407 
2504 
3601 
4698 

5796 
6894 
7992 
9090 
0188 

1287 
2386 
3485 
4584 
5684 
6784 


riff. 


log/',  log/" 


086 
086 
086 
087 
087 

087 
087 
0S7 
088 
088 

088 
0S9 
0S9 
089 
089 

089 
090 
090 
090 
090 

090 
091 
091 
091 
092 

092 
092 
092 
092 
093 

093 

093  j 
093 

094  I 
094  j 

094  I 
C95  I 
094 
095 
095 

095 
096 
096 
096 
096 

097    I 

097    ' 
097    : 

097  \ 

098  - 

098    I 
098   I 

098  I 

098  j 

099  I 

099 
099 
099 

100 
100 


0.301 

.301 

.301 
.301 
.301 

0.301 

.301 

.301 
.301 

.301 
0.301 

.301 

.302 
.302 
.302 

0.302 
.302 
.302 
.302 
.302 

0.302 

.302 

.302 

.303 
•303 

0.303 

.303 
.303 

•303 
.303 

©•303 

•303 
.303 

•303 
.303 

0.304 
.304 
.304 
.304 
•304 

0.304 

•304 
.304 
.304 
.304 

0.304 
.305 
.305 

•305 
.305 

0.305 
.305 
.305 
.305 
.305 

0.305 
.305 
.306 
.306 
.306 
.306 


Diff. 


0300 
1  169 

2037 
2906 
3776 

4645 
55»5 
6384 

7254 
8124 

8995 
9865 

0736 

1606 

2477 

3348 
4220 
5091 
5963 
6835 

7707 
8579 
945' 
0324 
1 196 

2069 
2942 

3815 
4689 

5562 

6436  I 
7310 
8184  i 
9058  i 

9933  I 

I 
0807 
1682  \ 
2557  , 
3432 
4308  1 

5183 
6059 
6935 

7811 
8687 

9563 
0440 

1317 
2194 

3071 

3948 
4825 

5703 
6581 

7459 

8337 
9215 
0094 
0973 
1851 
2730 


869 
868 
869 

870 

869  ; 

870 
869 
870 
87c 
871 

870 
871 
870 
871 
871 

872 
871 
872 
872 
872 

872 
872 
873 
872 
873 

873 
873 
874 

874 

874 
874 
874 
875 
874 

875 
875 
875 

876 

875 

876 
876 
876 
876 
876 

877 
877 
877 
877 
877 

877 
878 
878 

878 
878 

878 

879 
879 

878 

879 


oyy 


TABLE  XVII. 

For  special  Perturbations. 


q.  q',  q" 


0.0060 
.0061 
.006a 
.0063" 
.0064 

0,0065 
.0066 
.0067 
.0068 
.0069 

o  0070 

."lo-ri 
.0-072 
o<.73 
.coy4 

0.0075 
.0076 
.0077 
.0078 
.0079 

0.00?0 

.0081 

.0082 
.0083 
.0084 

0.0085 

.00S6 

.0087 

.00S8 

.0089 

0.0090 
.0091 
.0092 

.0093 
.0094 

0.0095 
.0096 

.0097 

.0098 
.0099 

O.OIOO 
.0101 
.0102 

.0103 
.0104 

0.0105 
.0106 
.0107 
.0108 
.0109 

O.OIIO 

.oil  I 

.0112 

.01 13 
.0114 

0.0II5 
.0116 
.0117 

.0118 

.0119 
.0120 


For  iiositive  values  of  tliu  Argument. 


log/ 


Diff.         log/',  log/"      Diff. 


0.470  6488 
.470   5416 

•470  4545 

-  .470  3274 

.470  2103 

0.470  1 1 32 

.470  0062 

.469  8992 

.469  7922 

.469  68 5 2 


0.469 
.469 
.469 
.469 
.469 

0.469 
.4')8 
.4(8 
.408 
.468 

0.468 
.468 
.468 
.468 
.468 

0.467 
.467 
.467 
.467 
.467 

0.467 
.467 
.467 
.467 
.467 

0.466 
.466 
.466 
.466 
.466 

0.466 
.466 
.466 
.466 
.465 

0.465 
.465 
.465 
.465 
.465 

0.465 
.465 
.465 
.465 
,464 

0.464 
.464 
.464 
.464 
.464 
.464 


5782 
4713 
3644 
^575 
1506 

0437 
9369 
8301 

7233 
6165 

5098 
4031 
2964 
1897 
0831 

976? 
8699 

7633 
6567 
5502 

4437 
3372 
2307 
1243 
0179 

9115 

8051 
69S8 

5925 
4862 

3799 
2736 
1674 
0612 
955° 
8488 

7427 
6366 

53°5 
4244 

3'83 
2123 
1063 
0003 
8943 

7884 
6825 
5766 

4707 
3648 
2590 


1072 
1071 
1071 
1071 
1071 

1070 
1070 
1070 
1070 
1070 

1069 
1069 
1069 
1069 
1069 

1068 
1068 
1068 
1068 
1067 

1067 
1067 
1067 
1066 
1066 

1066 
1066 
1066 
1065 
1065 

1065 
1065 
1064 
1064 
1064 

1064 
1063 
1063 
1063 
1063 

1063 
1062 
1062 
1062 
1062 

1061 
1 06 1 
1 06 1 
1061 
106 1 

1060 
1060 
1060 
1060 
1059 

1059 
1059 

«°59 
1059 

1058 


0.295  8495 
.295  7637 
.295  6779 
.295  5921 
.295  5063 


0.295 

.29  T 
.295 
.295 
.295 

0.294 
.294 
.294 
.294 
.294 

0.294 
.294 
.294 
.294 
.294 

0.294 
.294 

•293 
.293 
.293 

0.293 
.293 
.293 
.293 
.293 

0.293 

•293 
.293 

•293 
.292 

0.292 
.292 
.292 
.292 
.292 

0.292 
.292 
.292 
.292 
.292 

0.292 
.291 
.291 
.291 
.291 

0.291 
.291 
.291 
.291 
.291 

0.291 
.291 
.290 
.290 
.290 
.290 


4205 
3348 

349 « 
1634 

0777 

9920 
9064 

8208 
7351 
6495 
5640 
4784 
3928 

3073 
2218 

1363 

0508 

9653 
8799 

7945 
7090 
6236 
53*^3 
4529 
3675 

2822 
1969 
1 1 16 
0263 
941 1 

8558 
7706 
6854 
6002 
5150 

4298 

3447 
2595 
«744 
0893 

0043 
9192 
«34i 
749 « 
6641 

5791 
4941 
4092 
3242 
2393 

1544 
0695 

9846 

8997 

8149 

7300 


858 
858 
858 
858 
J858 

^857 

857 

:857 

I  857 

i857 

i856 
856 

857 
856 

855 

i  856 

I  856 

^55 

855 

I  855 

^855 

;855 

854 

;  -^54 

I  855 

i854 
853 
854 

854 

1853 

:853 
853 
853 
852 

1853 

'852 
852 
852 
852 


851 

852 
851 
851 
850 

851 
85, 
850 
850 
850 

850 

849 
850 
849 
849 

849 

849 
849 

848 
849 


For  negative  values  of  the  Argmnont. 


log/ 


0.483 
.483 

•483 
.484 

.484 

0.484 
.484 
.484 
.484 
.484 

0.484 
.484 
•485 
.485 

•485 

0.485 

•485 
.485 
.485 
.485 

0.485 
.485 
.486 
.486 
.486 

0.486 
.486 
.486 
.486 
.486 

0.486 

.487 
•487 
.487 
.487 

0.487 
.487 
.487 
.487 
.487 

0.488 
.488 
.488 
.488 
.488 

0.488 
.488 
.488 
.488 
.489 

0.480 

•4'9 
.489 

•489 
.489 

0.489 
.489 

■489 
.490 
.490 
.490 


Diff. 


6784 
7884 
8984 
0085 
I186 

2287 
3388 
4490 

5592 
6694 

7796 
8898 
0001 
1 1 04 

2207 

33" 

4415 
5519 
6623 

7728 

8833 
9938 
1043 
2149 
3255 
4361 

5467 
6573 
7680 
8787 

9894 

lOOI 

2109 

3217 
4325 

5433 
6542 
7651 
8760 
9869 

0979 
2089 
3199 

43°9 
5420 

653' 
7642 

8753 
9865 
0977 

2089 
3201 
4314 
5427 
6540 

7653 

8767 
9881 

°995  ! 
2109  I 

3223 


lOT 
100 
lOI 
lOI 
lOI 

lOI 

102 
102 

102 
102 

102 

103 
103 

103 
104 

104 

104 

104 
105 

105 

105 
105 

106 
106 
106 

106 

06 
107 

107 
107 

107 
108 
108 
108 
108 

109 
109 
109 
109 
no 

no 
no 
no 
ni 
m 

1 1 1 
ni 
n2 
n2 

1 12 

1 12 
113 
"3 
»i3 
113 

"4 
«i4 
'H 
i'4 
114 


log/',  log/" 


0.306 
.306 

.306 

.306 
.306 

0.306 
.306 
.306 
.306 

.307 

0.307 

•  3°7 
.307 
.307 
.307 

0.307 
.307 
.307 
.307 
.307 

0.308 

.308 
.308 
.308 
.308 

0.308 
.308 
.308 
.308 
.308 

0.308 
.309 
.309 
.309 
•3°9 

0.309 
.309 
.309 
.309 
.309 

0.309 
•309 
•309 
.310 
.310 


2730 
3610 
4489 

5369 
6248 

7128 
8009 
8889 
9769 
0650 

1531 

2412 
3293 

4J74 
5056 

5938 

6820 
7702 
8584 
9466 

0349 
1232 
2n5 

2998 
3881 

4765 
5648 
6532 
7416 
8301 

9185 

0070 

0954 
1S39 

2725 

3610 

4495 
5381 
6267 

7153 

8039 
8926 
981a 
0699 
1586 


q.  q'. 


0.310  2473 

.310  3360 

.310  4248 

.3IC  5136 

.310  6023 

0.310  6911 
.310  7800 

.310  8688 
.310  9577 
•3J'  0465 


0.3M 
•3" 
•3" 
•3»« 
•3" 
•3«" 


«354 
2243 
3'33 

4022 

49 '2 
5802 


Diff. 


880 ; 

879 

880 

8-^9 
880 

881 
880 
880 
881 
881 

88i 
881 
881 
882 
882 

882 
882 
882 
882 
883 

883 
883 
8S3 
883 
884 

883 

884 
884 
885 
884 

885  ' 

8,84 

885 

886 

885 

885 
886 
886 
886 
886 

887 
SS6 
8S7 
S87 
887 

887 
888 
888 
887 
888 

889 
888 
889 
888 
S89 

889 
890 
889 
890 
890 


634 


TABLE  XVII. 

For  special  Pertiirl)ation.s. 


3nt. 

tf 

Diff. 

0 

88o  i 

o 
9 

879  1 

880  ! 

I 

8-5 

880 

8 

881 

y 

880 

9 

8S0 

y 

881 

881 

I 

881 

881 

!> 

881 

t 

882 

882 

8 

882 

.0 

882 

)2 

4 
)6 

8S2 
882 

883  , 

1-9 
!2 

883  i 
881 

i 

883 

<i 

883 
884 

'i 

■  883  ' 

!  884  i 

11 

■  884 

>i 

885 
'  884 

885  ' 
884 

)4 

'  885 

(9 

1  886 

^S 

!  885 

10 

885 
886 

886 

'/ 

8S6 

>J 

886 

!9 

i6 

SS7 
SS6 

SSy 

S87  ■ 

887 

)0 

[8 
■3 

887 
888 

888 

887  , 

888  1 

I 

889 

!8 
'7 

i  888  1 
:  889 

i  888 

'i 

889 

14 
■3 
13 

,2 

i  889 
1  890 
!  889 

2 
)2 

1  890 
;  890 

0.0 
.0 
.0 

.0 

.0 

0,0 
.0 
.0 

.0 

.0 

0.0 

.0 

.0 

.0 
.0 

0.0 
.0 
.0 

.0 
.0 

0.0 
.0 

.0 

.0 

.0 

0.0 

.0 

.0 
.0 

.0 

0.0 
.r 

.0 
.0 
.0 

0.0 

.0 
.0 
.0 
.0 

0.0 
.0 
.0 
.0 

.0 

0.0 

.0 
.0 
.0 

.0 

I  o  o 
.0 

.0 


For  positive  viiliiea  of  the  Argument. 


20 
21 
22 

23 
24 

^5 
26 

27 
28 
29 

30 
3' 
3^ 
33 
34 

35 
36 
37 
38 
39 

40 
4' 
42 
43 
44 

45 

46  ; 

+7  i 
48 

49 

CO  I 

51 

52 

53 

54 

55 
56 
57 
58 
59 
60 
61 
62 

63  , 
64 

65 
66 

67 
68 
69 

70 
71 

71 


.0 

73 

.0 

74 

0.0 

75 

.0 

76 

.0 

77 

.0 

78 

.0 

179 

.0 

180 

log/ 


Dlff.     I     log/',  Ing/"    I    DIIT. 


0.464 
.464 
.464 
.463 
.463 

0.463 
.463 
.463 

•463 
.463 

0.463 
.463 
.462 
.462 
.462 

0.462 

.46  ^ 

.46-... 
.462 
.462 

0.462 
.462 
.461 
.46, 
.461 

0.461 
.46, 
.461 
.46: 
,461 

0.46 1 
.460 
.460 
.460 
.460 

0.460 
.460 
.460 
.460 
.460 

0.460 

•45.' 
•459 
•459 
•459 
0.459 
•459 
■459 
•459 
•459 

0.458 
.458 
.458 

•458 
.458 

0.458 
.458 
.458 
.458 
•458 
•457 


2590 
1532 

°474 
9416 

8359 

7302 
6245 
5188 
4132 
3076 

2020 
0964 
9908 

8853 
779« 

6743 
';688 

4633 
3579 


1471 
0417 
9364 
8311 
7258 

6205 

5«53 
4101 
3049 
»997 

0945 
9894 

8843 

779i  , 
6741 

5690 
4640 
3590 
2540 
1490 

0441 
9392 

8343 
7194 
6245  1 

5197  I 
4149 
3101 
2053  I 
1006  j 

9959  i 
8912 

7865  i 
6818 

577i  I 

4726  ' 

3680  j 

2634 

1589 

0544 

9499 


058 
058 
058 

057 
057 

057 

056 
056 
056 

056 


055 
055 
055 

055 

°55 
054 

-154 
054 

"54 
053 
°53 
°5  3 
053 

052 
052 
052 
052 
052 

051 
051 
051 
051 
051 

050 

050 
5<: 


050 
050 
049 

049 
049 
049 
049 
048 

048 
048 
048 
047 

047 
047 
047 
046 
046 

046 
0415 

045 
045 

°4S 


0.290 
.290 
.290 
.290 
.290 

0.290 
.290 
.290 
.290 
.289 

0.289 
.289 
.289 
.289 
.289 

0.289 
.2S9 
.289 
.289 
.289 

0.289 
.288 
.288 
.288 
.288 

0.288 
.288 
.288 
.288 
.288 

0.288 
.288 
.288 
.287 
.287 

0.287 
.287 
.287 
.287 
.287 

0.287 
.287 
.287 
.287 
.287 

0.286 
.286 
.286 
.286 
.286 

0.286 
.286 
.286 
.286 
.286 

0.286 
.286 
.285 
.285 
.285 
.285 


7300 
6452 
5604 

4756 
3909 

3061 
2214 
1367 
0520 
9673 

8826 
7980 

7134 
6287 

544> 

4596 

3750 
2904 
2059 
1214 

0369 

95^4 
8679 

7835 
6990 

6146 

5302 

4458 
3615 

"•77' 

1928 
1085 
0241 

9399 
8556 

7713 
6871 

6029 

5187 

4345 

3503 
2661 
1820 
0979 
0138 

9297 
8+56 
7615 
6775 
S93S 

5095 
4155 
3415 
^575 
1736 

0896 
0057 
9218 
8380 

754« 
6702 


848 
848 
848 

847 
848 

'847 
847 
847 
847 
847 

'  846 
846 

847 
'  846 

■845 

'  846 

846 

845 

,845 

'845 

845 
,  S45 
844 
845 
844 

■844 
:  844 

i843 
844 

1843 

843 
844 

842 

843 

1843 

842 
842 
,842 
842 
842 

,842 
'  841 

84. 

84. 

841 

84. 

841 

840 

,  840 

1  840 

840 
840 
840 
839 
1  840 

;839 
839 
838 
839 
839 


For  nog-itive  values  of  the  Argument. 


los  / 


Dlff. 


0.490 

•49° 
.490 

■49° 
.490 

490 
490 

49' 
49' 
491 

491 
491 
491 
49' 
491 

49' 

492 
492 
492 
492 

492 
492 
492 
492 
493 

493 
493 
493 
495 
493 

493 
493 
493 
494 
494 

494 
494 
494 
494 
494 

494 
494 
495 
405 

495 

495 
495 
495 
495 
495 

495 
496 
496 
496 
496 

496 
496 
496 
496 
496 
497 


3223 
4338 

545  3 
6568 
7684 

8800 
9916 
1032 
2149 
3266 

4383 
5500 
6618 
7736 
8854 

9972 
1 09 1 

2210 

3  3^9 
4448 

5567 
6687 

7807 
8927 
0047 

1168 

22>;9 
3410 
453- 
5<^54 
6776 
7898 
9021 
0144 
1267 

2390 

35'4 
4638 
5762 
6886 

8010 

913s 

0260 
1385 
2510 

3616 

4762 
588S 
7015 
8142 

9269 
0396 
1524 
2652 

3780 

4908 
6037 
7166 
8295 
9424 
055+ 


I 
log/',  log/"    j  I'iff- 


0 

I1I5 
1115 

1115 
11 16 

III6 

11,6   ° 

1116 

1117 
M17 
H17 

1  0 

1117  1 

1118  ! 

1118 

1118 

1118 

0 

1119 
1119 

1119 

1 1 19  ! 

MI9 

0 

1120 

1120 

1120 

1120 

II2I 

0 

iiai 

1121 

1122 

1122 

1122 

0 

1122 

1123 

1123 

1123 

1123 

0 

1124 

1124 

1124 

1124 

1124 

0 

1125 

1125 

1125 

1125 

1126 

.126    '= 

1126 

1127 

1127 

1127 

0 

II27 

1128 

1128    . 

1128 

1128 

0 

1129 

1129 

1129 

31 

1129 

1130 

I   5802 

1  6692 

1  7582 
1  8472 
1  9363 

2  0254 
2  I  145 

2  2036 

2  2927 

2  3S19 

2  4710 
2  5602 
2  6494 
2  73S7 

2  8279 

3  9172 

3  0064 
3  °957 
3  '850 
3  2744 


3637 
453' 
54^5 
6319 
7213 

13  8108 

13  9002 

13  9897 

14  0792 
14  1687 

14  2583 
'4  3478 
'4  4374 
'4  5*7° 
14  6166 


7062 

7959 
8855 

97  5  2 
0649 


5  1546 
5  i444 
5  334' 
5  4239 

5  5' 37 

5  6035 

5  6934 

5  7832 

5  873' 

5  9630 


0529 
1428 
2327 
3227 
4127 

5027 

5927 
6827 
7728 
8629 
9530 


;  890 
S90 
890 
891 
891 

891 
891 
891 
891 
,  891 

892 

892 

893 

892 
i893 
\  892 
,893 

893 

894 
,893 

894 

'894 
894 

894 
;89S 

1894 
1 895 
'895 
.895 
896 

895 

:  896 

896 

896 

i  896 

897 
.  896 

897 

1897 
897 

,  89S 

897 

898 
,898 

898 

1899 

■■   898 

899 
,899 

■  899 

899 

,899 
900 

900 
I  900 

I  900  i 

900 

goj 
.  901 
'  901 


I 


635 


si 

i 
n 


TABLE  XVII. 

For  special  Perturbations. 


1.  9'.  7" 


0.0180 
.0181 
.01X2 
.0183 
.0184 

0.0185 
.0186 
.0187 
.0188 
.0189 

0.0190 
.0191 
.0191 
.0193 
.0194 

1  0.0195 

i  .0196 

i  -oi'jy 

.0198 

.0199 

1  0.0200 
!  .0201 
]  .0202 
I  .0203 
.0204 

0.0205 
.0206 

.02Q7 
.0208 
.0209 

0.0210 

'   .02II 

.0212 

.0213 

.0214 

0.0215 
.0216 
.0217 
.0218 
.0219 

0.022n 

.0221 

.0222 
.0223 
.0224 

0.0225 
.0226 
.0227 
.0228 
.0229 

0.0230 
.0231 
.0232 
.0233 
.0234 

0.0235 
.0236 
.0237 
.0238 
.0239 
.0240 


For  positive  values  of  the  Argument. 


log/ 


0-457  9499 
•+57  »454 
•457  7409 
•457  6365 
■457  S3J> 


457 
457 
457 
457 
457 

456 
456 
456 
456 
456 

456 
456 
456 
456 
455 

455 
455 
455 
455 
455 

455 
455 
455 
455 
454 

454 
454 
454 
454 
454 

454 
454 
454 
453 
453 

453 
453 
45  3 
453 
453 

453 
453 
453 
452 
452 

452 

452 
452 
452 
452 

452 

45* 
452 

45 « 
451 
4SI 


4177 

3*33 
2189 

1 146 

0103 

9060 
801/ 
6975 

593  J 
4891 

3849 
2808 
1767 
0726 
9685 

8644 
7604 
6564 

5524 
4484 

3444 
2405 
1366 
0327 
9288 

8249 
7211 

6173 

5135 

4097 

3060 
20 -.'.3 
0986 

9949 
8912 

7876 
6840 
5804 
4768 
3733 
2698 
1663 
0628 

9593 
8558 

7514 
6490 

5456 
4422 
3389 

2356 
1323 
0290 
9258 
8226 
7»94 


Diff. 


1045 
1045 
1044 
1044 
1044 

1044 
1044 
1043 
1043 

1043 

'°43 
1042 
1042 
1042 
1042 

1 041 
1 041 
1041 
1 041 
1041 

1040 
1040 
1040 
1040 
1040 

1039 
1039 
1039 
1039 
1039 

1038 
1038 
1038 
1038 
1037 

1037 
1037 
1037 
1037 
1036 

1036 
1036 
1036 

i"35 
1035 

1035 
1035 

1035 
1035 
1034 

1034 
1034 
1034 
1033 
»033 
1033 
1033 
1032 
1032 
1032 


iog.r,  log/" 


0.285 
.285 
.285 
.285 
.285 

0.285 
.285 
.285 
.285 
.284 

0.284 
.284 
.284 
.284 
.284 

0.284 
.284 
.284 
.284 
.284 

0.283 
.283 
.283 
.283 
.283 

0.283 
.283 
.283 
.283 
.283 

0.283 
.283 
.2S2 
.282 
.282 

0.282 
.282 
.282 
.282 
.282 

0.282 
.282 
.282 
.282 
.281 

0.281 
.281 
.281 
.281 
.281 

0.281 
.281 
.281 
.281 
.281 

0.281 
.281 
.280 
.280 
.280 
.280 


6702 
586A 
5026 
4188 
3350 
2512 
1675 
0838 
0000 
9163 

8326 
7490 
6653 
5817 
4981 

4«i5 

3309 

2473 
1637 
0802 

9967 
9132 
8297 
7462 
6627 

5793 
4958 
4124 
3290 
2456 

1623 

0789 
9956 
9123 
8290 

7457 
6624 

5792 
4959 

4127 

3295 
2463 
1631 
0800 
9968 

9137 

8306 

7475 
6644 
5814 

4983 
4«53 
3323 
2493 
1663 

0S33 
0004 
9174 
8345 
7516 
6687 


Diff. 


838 
838 
838 
838 
838 

1837 

i837 
1838 

837 
I  837 

:  836 
^37 
S36 

;  836 
I  836 

'  836 
836 

835 
'835 

:835 
;835 
1835 

i834 

:^35 

^54 

833 

833 
833 
833 
833 

833 
832 

!" 
832 

832 
832 

831 
832 
831 

831 
831 
831 
830 
831 

830 
830 
830 
830 
830 

829 
830 
829 
829 
829 

636 


For  negiilivc  viiluos  of  the  Argiiincnt. 


log/ 


0.497 
•497 
•497 
•497 
•497 

0.497 
•497 
•497 
•497 
.498 

0.498 
.498 

•498 
.498 

•498 

0.498 
.498 
.498 
•499 
•499 

0.499 
•499 
•499 
•499 
•499 

0.499 

.500 
.500 
.500 
.500 

0.500 
.500 
.500 
.500 
.500 


0554 
1684 
2814 
3944 
5°75 
6206 

7337 
8468 
9600 
0732 

i86j. 
2996 
4129 
5262 
6395 
7528 
86b2 
9796 
0930 
2064 

3199 
4334 
5469 
6604 

7740 
8876 
0012 
1 149 
2286 

3423 
4560 
5697 
6835 
7973 
9111 


Diff. 


0.501  0250 
.501  1389 
.501  2528 
.501  3667 
.501    4807 


5947 
70S7 
8227 
9368  ; 
0509  i 

1650 
2791  : 
3933 

5°75 
6217  I 

7360  ; 

8503  ; 
9646 
0789 
1932 

3076 
4220 

5364 
6508 

7653  i 
8798 


0.501 
.501 
.501 
.501 

.502 

0.502 
.502 
.502 
.502 
.502 

0.502 
.502 
.502 
.503 
.503 

0.503 
.503 
.503 
.503 
.503 
.503 


log/',  log/" 


130 
I  30 
130 

>3> 

131 

131 
131 

>3^' 
132 
132 

132 
133 
133 
133  ' 

133  ! 

>34  ! 

134  I 

'34  i 
134  I 
'35  i 

'35  I 
'35  ! 
'35  ! 
136 
136 

136 

'37 
'37 
'37 
'37  I 

.37 
'.III 

138  I 
'39  ! 

'39 
'39 
'39 

140  I 
140 

140 
140 

141  1 
'41  I 
.41  i 

141  ! 

142 

142  I 
142 

'43 

143 

'43 
'43 
'43 
'44 

'44 
'44 
'44 
145 
145 


0.3 
•3 
3 
3 


6  9530 

7  043' 
7  1332 
7  2234 
7  3'35 

7  4037 

7  4939 

7  584' 

7  6744 

7  7646 


Diff. 


7  8549 

7  9452 

8  0355 
8  1259 
8  2162 

8  3066 

8  3970 

8  4874 

8  5778 

8  5683 

8  7588 

8  8492 

8  9398 

9  0303 
9  1208 

9  2114 

9  3020 

9  3926 

9  4832 

9  5738 

9  6645 

9  7552 

9  8459 

9  9366 

320  0273 

320  ii8i 

320  20S8 

320  2996 

320  3904 

320  4813 

320 
320 
320 
320 
320 

321 

32' 
321 
321 
321 

321 
321 
321 
321 
32' 


5721 
6630 

7539 
8448 

9357 

0266 
1 176 
2086 
2996 
3906 

4816 

5727 
6637 

7548 
8460 


321  9371 

322  0282 
322  1194 
322  2106 
322  3018 
322  3930 


901 
901 
902 
901 
902 

902 
902 
903 
902 
903 

903 
903 
904 
903 
904 

904 

904 
904 
905 
905 

904 
906 
905 
905 
906 

906 
906 
906 
906 

907 

907 
907 
907 
907 
908 

907 
908 
908 
909 
908 

909 
909 
909 
909 
909 

910 
910 
910 
910 
910 

9" 

910 

9" 
912 
911 

911 
912 
912 
912 
912 


TABLE  XVII. 

For  special  I'erliirbutions. 


?.  '/'.  7" 


DilT. 

> 

1)01 

901 

Q02 

0.0240 
.0241 
.0242 

.0243 
.0244 

0.0245 

.0246 

.0247 
.0248 
.0249 

0.0250 

.0251 

.0252 

.0253 
.0254 

0.0255 
.0256 

.0257 
.0258 
.0259 

0.0260 
.0261 
.0262 
.0263 
.0264 

0.0265 
.0266 
.0267 

.0268 

.0269 

0.0270 
.0271 

.0272 

.0273 
.0274 

O.C275 
.0276 
.0277 
.0278 
.0279 

0.0280 
.0281 
.0282 
.0283 
.0284 

0.0285 
.0286 
.02S7 
.0288 
.0289 

0.0290 
.0291 
.0292 
.0293 
.0294 

0.0295 
.0296 
.0297 
.0298 
.0299 

.OJOO 


For  i)OsitiTO  vnliiea  of  the  Argument. 


log/ 


0.451  7194 
.451  6162 

45'  S'30 
451  4099 
451  3068 


Diff. 


log/',  log/"     Diir. 


451 
451 
450 
450 
450 

450 
450 

45° 
45° 
450 

45° 
450 
449 
449 
449 

449 
449 
449 
449 
449 

449 
449 
448 
448 
448 

448 
448 
448 
448 
448 

448 
448 
447 
447 
447 

447 
447 
447 
447 
447 

447 
446 
446 
446 
446 

446 
446 
446 
446 
446 

446 
445 
445 
445 
445 
445 


2037 
1006 

9975 
8945 
79«5 
6885 

5855 
4825 
3796 
2767 

1738 
0709 
9681 

8653 
7625 

6597 
5569 
4542 
3515 
2488 

1461 

°435 
9409 
8383 
7357 

6331 

5305  I 
4280  I 

3255  ' 
2230  i 

1205  I 
0181  I 
9'57 

8133  : 

7109  I 

6085 
5062 

4°39  : 
3016 

2993  j 

0970  1 

9948  I 
8926  I 
7904  ' 
6882 

581:1 
4840 

3819 
2798 

>777 

0756 

9736 
8716 
7696 
6676 
5657 


1032 
1032 
1031 
1031 
1031 

1031 
1031 

1030 
1030 
1030 

1030 
1030 
1029 
1029 
1029 

1029 

1028 
1028 
1028 
1028 

1028 
1027 
1027 
1027 
1027 

1026 
1026 
1026 
1026 
1026 

1026 

1025 
1025 
1025 
1025 

1024 
1024 
1024 
1024 
1024 

1023 
1023 
1023 
1023 
1023 

1022 
1022 
1022 
1022 
1021 

102 1 
1021  I 

I02I  ; 
1021 
1 02 1 

1020 

1020  I 
1020  j 
1020 

1019  ; 


0.280 
.280 
.280 
.280 
.280 

0.280 
.280 
.280 
.280 
.279 

0.279 
.279 
.279 
.279 
.279 

0.279  4273 
.279  3446 
.279  2620 
.279  1794 
.279  0968 


6687 
5858 
5030 
4201 
3373 
2545 

I7>7 
0889 
0062 
9^34 

8407 
7580 

6753 
5926 
5099 


0.279 

.278 
.278 
.278 
.278 

0.278 
.278 
.278 
.278 
.278 

0.278 
.278 
.278 

•277 
.277 

0.277 
.277 
.277 

•277 
.277 

0.277 
.277 
.277 
.277 
.277 

0.276 
.276 
.276 
.276 
.276 


0143 

9317 
8492 
7666 
6841 

6016 
5191 
4367 
3542 
2718 

1894 
1070 
0246 
9422 
8599 

7775 
6952 
6129 
5306 
4483 

3661 
2838 
2016 
1194 

0372 

9550 
8728 
7907 
7086 
6264 


o-27<'  5443 
.276  4622 
.276  3802 
.276  2981 
.276  2161 

0.276  1340 
.276  0520 
.275  9700 
.275  8880 
.275  8061 
.275  7241 


829 
828 
829 
828 
82S 

828 
82S 
827 
828 
827 

827 
827 
827 
827 
826 

827 
826 
826 
826 
825 

826 
825 
826 
825 
,825 

i  825 
824 
825 
824 

1824 

I  824 
824 

'S24 
823 
824 

823 

823 
823 

;823 
j  822 

!  823 
'  822 
822 
]  S22 
j  822 

822 

:  821 

i  821 

822 

821 

821 
I  820 
1  821 

820 

I  821 

!  820 
820 
820 
819 

820 


For  nugativo  vuluos  of  tiio  Argitiiioiit. 


\osf 


DifT.        log/',  log/" 


Pi  IT. 


0.503 
.503 
.504 
.504 
.504 

0.504 
.504 
.504 

•504 
.504 

0.505 
.505 
.505 
.505 
.505 

0.505 
.505 

•5°5 
.505 
.506 

0.506 
.506 
.506 
.506 
.506 

0.506 
.506 
.506 

.507 
.507 

0.507 
.507 
.507 
.507 
.507 

0.507 

.508 
.508 
.508 
.50S 

0.508 
.508 
.508 
.508 
.508 

0.509 
.509 
.509 
.509 
.509 

0.509 
.509 
.509 
.509 
.510 


8798 

9943 
1089 
2235 
3381 

4527 
5674 
6821 
7968 
91 15 

0263 
1411 

2559 
3707 
4856 

6005 

7154 
8303 

9453 

0603 

•753 
2903  ; 
4054 

6356  ■ 
7508  ] 

8660  ! 

9813  : 

0965  ; 
2117  i 

3^7°  : 
4423 

5577  , 
673'  I 

7885  ! 

9039  I 
0194  ' 

'349 

2504 

3659  , 

4814  i 
5970 
7126 
8282 
9439  : 
0596  ' 

1753  : 
2910  I 

4068  ; 
5226  j 

6384  ; 

7543  I 
8702  ' 
9861  '' 
1020 


0.510  2179 
.510  3339 
.510  4499 
.510  5659 
.510  6819 
.510  79S0 


45 
46 
46 
46 
46 

47 
47 
47 
47 
48 

48 
48 
48 
49 
49 

49 
49 

50 
50 

5° 

50 
51 
5' 
5' 


52 
53 
52 
52  I 
53 

53 
54 
54 
54 

54  I 

55  ' 
55  , 
55 
55 
55  ' 

II 

56 

57 

57 ; 

57 

58 1 
58  i 

59 

59 1 

59  ; 

59 

60 

60 
60 
60 

6i 


0.322 

•322 
.322 
.322 
.322 

322 
322 
323 
323 
323 

323 
323 

323 
323 
323 

323 
323 
323 

324 
324 

324 
324 
324 
324 
324 

324 
324 
324 
324 
325 

325 
325 
325 
325 
325 

325 
325 
325 
325 
325 
326 
326 
326 
326 
326 

326 
326 
326 

32fi 

326 
326 

327 
327 

327 
327 


3930 

4843 
5756 
6668 
7581 

8495 
9408 
0322 
1236 
2150 

3064 

3978 

4893 
5808 

6723 
7638 

8553 
9469 

0384 

I  300 

2217 

3133 
4049 
4966 
5883 

6800 

7717 
S635 

9553 

0470 

1389 
2507 
3225 
4144 
5063 

5982 
6901 

7821 
8740 
9660 

05  So 
1500 
2421 

3341 
4262 

5183 
6104 

7026 

7947 
S869 

9791 
0713 

'635 
2558 

3481 


327  4404 
327  5327 
327  6250 

327  7'74 
327  8097 
327  9021 


920 
920 

920 
921 
920 
921 
921 

921 
922 
921 
922 
922 

922 
922 
923 
923 
923 

923 
923 
924 
923 

924 


637 


TABLE  XVIII. 

Elements  of  the  Orbits  of  Comets  wliicli  have  been  observed. 


] 

3 

a. 
S 

5 

'H       -H 

^   =* 

^ 

J3                c5 

.     b4             t4 

U        .     b4 

►-•  ..s 

O 

0)  >i; 

-2   '1  to-i 

k5  '«S 

■  'i.2  a;  .2 

-d  'u  So'tc  ^ 

ii-m 

Nill 

S'^M 

K  *  « •  =  - 

Hi          M  l-H  »-H 

xt^i::^ 

SasJ 

K^^K^ 

oxzi^-i 

V 

V           a! 

« 

6 

TS                -o 

T3 

-B             "O 

rs 

.2 

S                S 

2 

2         2 

£ 

2 

g 

*:      tc 

^  ..  - 

&•>  t*  iP 

^    ^    ^    ^ 

■*J   50  .^    .^  -*-! 

tl3^  ^ 

-  -  j^ 

£'  1  £ 

S 

a     Qj     c 

O           b^ 

O    I-                 K 

>-        o 

■K                 k-      t      -f 

i«  s  -W 

E  &  & 

^  aJ            ^ 

a  a  -r,       t- 

(S    5    « 

s   « 

a 

~l  s« 

Ss;      s 

X    5 

^00 

00 

in  o    *■ 

1 
tl             1 

e< 

t-^VO 

«3   r> 

H 

to  in  o    ^ 

O   ^  ^  o  -^ 

so  OO    *♦■ 

so    to  0    O    tl    ' 

0    m  o    o  00 

N.  so  VO    tJ-00 

\o  toso  to  r^ 

00     to  w    ^   O 

vO    t^  ^r\  o    r^ 

OO    O    m  to  X 

to  tl     ^   tl  00 

tc 

00   ^~  o   t^vD 

Cn  »n  r^  m  .- 

^  t^  t^oo  so 

to  H   OO     •-  SO 

oc    0    t--  *-•    « 

1^00  so    to  f- 

T*-   fO  .^     OS  O. 

tJ-  in  r-,  t*^  ir\ 

•>l-oe    o  ^   ■>*• 

in  mso  vo   r^ 

OS  O     -«  OO  SO 

OO    (^\0    r*^  vj-i 

so     1^00  SO   OC 

«-    O  SO    >1-  1^ 

vo  00   u-1  inoo 

O  ONOO    1^  t^ 

1-^00     OOO     OS 

Th  in  OS  OS  t^ 

oc   U-*  r~-  o  r-- 

oc  00   m  t^  t^ 

m  r~  r--  tl   r^  j 

C\  Ov  0\  0\  Ov 

CT^  ^  ^  ^  ^ 

OS  as  OS  OS  o 

Os  OS  OS  ON  OS 

Os  O^  0^  ^  On 

Os  Os  OS  OS  OS 

Os  ON  Os  Os  cK   ' 

i 

M 
OS 

to 

u 

to 

SO    1 

oc 

Os 

c  0.  q  q  q 

q  q  q  q  q 

q  q  q  q  q 

q  q  q  q  q 

q  q  q  q  q 

M      )•«      M      H     M 

0  q  q  OS  q 
>"■  •-■  •-■  d  -" 

q  q  q  q  ov  1 

M      M      M       M      0       ' 

t    0    0    0   O    0 

o  o  o  o  0 

0   0   0   0   0 

O    0   0    0    0 

O    0    0   0    o 

0    0    0    0    0 

O    0    O    rt    m 
■<*■  m  1 

^    O   0   0   o   o 

rl   w    CT\  0    to 

0    O    ^n  O    in 

m  t^oo   0  so 

m  n  so    OS  in 

t~-  0    1    o    t^ 

Th  tl    OS  OS  to 

m 

TO  ^        m 

in  to 

in  H         m 

M         in  NN    m 

to                   tl 

11     11     tl            to 

0  ^rr$2.2 

rj-vo    «    «    On 

r^  t^oo   fo-o 

moo    O  so    t-- 

tl   OS  r^  *  « 
in  t^  n   tJ- 

w    in  m  t^  tl 

00    O    to  in  **-   ' 

'l-vo   >-   r^ 

M    w    tl    1-^ 

toso    •>*•          M 

m  t-  ■♦  n    •* 

tl  to  r^  t^sc 

0 
5;     O     0     0  00     0 

O    O    O    O    0 

0   0   0   0   0 

0    O    O    O    O 

O   O    0    0    o 

0    0   0         0 

1 

0    0    O    'J-  m  1 

fl 

tl    tl 

V     0    O    0    I.    0 

vO    t-^  ^  ro  in 

0  0  o  o  o 

O  00    M     0     t-^ 

w   OS  0   m  tl 

m  0    C     5  00 

OssO  so    O    1-^    J 

a 

•i-  ^r^         O 

fO  h-    »n  to  ro 

in  rr  to  to 

'^ 

to   T^    to   M      to 

tJ»         ir    .-o 

NN      tl       to    tl 

M    rt    OVOO  00 

•<»-00    O  so    O 

^  m  in  t^  to 

t^  t^  to  f»    1^ 

OO  00    tl    in  OS 

Os  in  tl    in  Os 

0     m  «  00    ir^  "^ 

O-  rt    0\  0    m 

00   t<   ti   o    M 

mo    OS  >-   ■<*• 

so    to  ^so    O 

00  so     to  -^  -. 

OS  t^  to  tl    « 

-          - 

t<    -          D    to 

«   t« 

»   ►.         tl 

tl    «                tl 

tl    H    - 

N    1-.    to 

i     0    0    0    0    0 

O    0    0    0    0 

0    0    0    0    0 

O    O    0    O    O 

O    0    O    O    0 

O    0    O    O    0 

O    O    O    0    0 

»     0    "^  0    0    0 

t^  On  t^  to  to 

0   in  0    0  00 

00    0    O    0    M 

t^  tl    0    to  to 

Q    0    t^  tl    i 
<*■         to  -    .^ 

O    m  Os  tl    OS 

t: 

to        to 

^  to 

in  e)    to  tJ- 

to  tl    tl          to 

■♦ 

•<J-  «    ■*  't-  tl 

in  M    M    r^^oo 

VO     to  t^   ^00 

•+  tJ-so    t<    ■<♦• 

»i    to  f»  so    Os 

««    X    ■<  so  00 

OO    to  0    **    I's 

r--  ■+  Os  osoc 

o    f  1   »ri  r^  i-i  00 

«   rh  inoo  ^ 

so  so    m  to  to 

•+               so    ON 

O  OO   O    m  ^ 

m  **   in  o    to 

•>    t^  tl    tl    0    i 

m  r»    f»    m 

to  w    to  tl    H 

tl    f<    M    to  « 

M                        tl 

«    tl    to  to 

n   tl   to  ~ 

rl   rl    to  w  IN    ; 

»  O    O    0    0    O 

CT\  •*  >«    0    "^ 

O    0    0    OS  O 

•    0    O    O    t^ 

m  w    0    0    to 

O   0   •♦  0    0 

in  0    inso    Os  1 

■+  -                  tl 

tOTj- 

tl 

tl    m  Tf-  *  -. 

in        to  ^ 

tl         Tt*  to  to 

g  O    0    0    0    0 

00     tOVD     0     0 

0    0    O  SO    tl 

M    Os-«    0    OS 

T»-  OS  0    OS  to 

so    m  tl    0    to 

w   m  -^  •*  so 

M   ■+             m 

«    m 

to  tl    ■*          m 

M                        ■+    -. 

M    to  m        in 

X   tl    rl   •+ 

«  in  rt   0   in  0 

vo  vo    ■+  0    to 

0    0    0    «    t-~ 

to  t-  n    0  00 

so    0    tl    OS  m 

«   M   m  OS  T^ 

x   O   tl   tl   to 

E^ 

M 

rt 

m                        M 

M    H 

»    tl    n    M    « 

n        n   rl  " 

•^  05  ©"o  o 

00  t-  «o  —  o 

(N  r-H  lO  r-(  © 

5^  d  lO  CO  00 

to  ■^  00  l-»  00 

f  f  CO  lO  03 

"*  Ol  ©  to  «: 

i-l  (M  r-l  (M 

IT)                   CO 

I-l          1-1  Oq  CO 

11  CO  n  tl 

r-1                       «^ 

5-1                   Ul  r^ 

11  01  n  Ol  01 

Jan. 

March 

Nov. 

Oct. 

July 

Aug. 

April 

.June 

March 

Dec. 

Sept. 

April 

Feb. 

Sept. 

Jan. 

July 

March 

June 

Oct. 

Nov. 

Oct. 

Nov. 

June 

Oct. 

Feb. 

Dec. 
.Jan. 
Sept. 
Aug. 
Oct. 

CO  r-(  O  Ci  lO 

00  ■»f  ©  1^  'H 

05  «0  ■M  t^  ^ 

-i<  05 1^  to  00 

lO  CO  to  00  Ol 

O  n  to  d  01 

CO  to  CC  1-  o 

O  "t  -rf"  M  to 

«C  1^  f^  CO  -o 

00  -^  O  C5  CO 

to©co-o  l~ 

00  CO  10  to  I~ 

c;  o:  o  CO  CO 

CO  lO  lO  1-  X 

rH  (M  IC  UO 

lO  lO  1-  CC  05 

o;  o  o  ©  c-i 

IM  C-l  CO  CO  CO 

CO  If  tl  -f<  It* 

■^  -*!  lO  IC  o 

lO  >C  iC  lO  10 

f— 1  T-1  d  i—t 

f-H   1-1   1-H   1-1   ^1 

^^  '"*'"''"'  ^" 

iH»ieo^»9 

«et>.x8i© 

■F^  ^IM  ^  i» 

^r-oCCi© 

-"»1  W  •*  19 

«e  1-  GT  St  © 

iM  91  OS  •»»<  »9 

^. 

TH 

^NrHFHnr^ 

^H  ^  rH  ^N  91 

»l  »1  ®1 91 91 

91 91 91 91  99 

W  50  ©•  CO  so 

088 


TABLE  XVIII. 

Elements  of  the  Orbits  of  Comets  which  have  been  observed. 


^  ■-■£,  S'  -r' 


o  =:   v 

"S    3 


so   r^  o   o   n 
fi  rl    Tt-  rl  00 

»  o  ^  ■+ 
un  r--^  r-^  fi_  r^ 

o\  o^  d^  o^  o 


I- 


O    O    O    CT^ 

»  »'  ><'  d 


o  o  o 


^  rt  o^  a\  t^ 
M   M   r1         f*^ 


t*^  u-t  ^ 

I  t^  t^>o 


o  o 


CN  r^  ^*^  f*    ■" 


O    O    O    O    O 

O    ^r\  0\  t\    On 
^  «    ^  ^  fl 

f~-  ■<}-  ON  OX 

..   r-  rl   rl   O 
fi   rt    r»i  M   1- 


u-)  O  »nvo   O^ 

rl  ^  d  r^ 

►.    r)  rt    •* 

».    O  M    H    r" 

r)  «    cl    - 


rt  Cl  1—1  (M  •  • 

a  CU  =  o  ,-- 


1     cc  ?o  CC  1-  ® 
:    CO  ic-  ic  1^  * 
r    lo  ic  »c  1^  »*  1 

.i      „  r-l  .-(  rl  -- 

J    ,-<  (Jl  M  i»»< »« 
9    MMOeOSM 

1 

3 

c 
S 
C 

D'Arre-^t. 

Peters  and  Sawitsch. 

Hind. 

La  Caille. 

Hind.                               1 

Bessel.                            1 

Pingre. 

Bessel. 

Halley. 

Meehain. 

Lindelof. 
Halley. 
Henderson. 
Halley. 

'C       jj  _• 

o  -  0  J-  Ji 

Vogel. 
Burckhardt. 
ILUlev. 
La  Caille. 

] 
i    1 

c!       oj       oS 

na        T3        T3 

2      2      2 

£  j;  2  g  2 

s''' 

1     i 

1  s'l 

,^3  ^   ^    oS 

1 

«5< 

tc 

vo    I10O    O    rl 
m  m  r*^  tJ-  o 
».    r^  -c    o\  r- 

VO    On  ■«*■   O  fO 
rl    m  in  *t-  'n 
rl    0    t-~  C7.  t-^ 
^  d    CTsOO    CT\ 

00      O    ^      O      "H 
U^    0      W-i   *♦■    Tn 

CTn  O    O  CC  vO 
^    -^    O  rl    ^ 
r^  t^  u-»  owo 

0^  0^  ^  Cn  C^ 

ON  OS       NO    rl 
»1-  0    O    t--  t^ 
On  to  ON  •>»-  O 
O    1^   On  tooo 
"•    rl    ON  •+  ■<*• 
q    0_    tooo    + 
0    On  On  ON  On 

<-i  oo  00 
ON   1*^   ^   ^    ON 

i-i    'noo    ^    ro 
On  ON  *^  O    '"^ 
OC    ro  «ri  fo  rl 
vn  o^-O    tJ-oc 

0^     t;-   r-;-    t^    CTv 

O   t^  0^  ON  cS 

CO              0    0 

00     O           NO     t^ 

00   CI   w  NO   »n 

n    r^NO    On  *. 

11    r^  rl   to  r^ 
in  rl   ONOO  00 

of>oo'  c^  <r>  rfi 

O    M  OO  OO 
00    ON  0\^    o 
t-.  t--  rl    fo  On 
t\    0    O    '^  o 
r-^  t-i   fo  ro  "i* 

t^OO  NO    On  0 

CN  ON  o*  o»  0 

to  0 

"J-  t--  O           00 

t-.  «nNO    rl  00     1 

*♦•    I  •-    ON    O      CO     j 

CnO    t-cc  CO    ; 
o»  o    •+•  to  rl    1 
a\^_    to  c^00    j 

On  d    C^  O^N  ^ 

V 

« 

q  q  q  q  q 

00 

o 

NO 

ON  0    O    0    o 

d  «■  -.'  «■  «■ 

q  q  q  q  q 

«  o 
^  «   0 
O    »^  O    1^ 
t^oo    r»    ^ 

ON  On  Ov  Cl 

o   ON  r-  to 

f*     ON*^  OO 

NO  o>  o\  ON  q 
d  d  d  d  w 

q  q  q  q  q 

q  q  q  q  q 

ro 
to 

o 

«n 

o 

q  q  q  q 

•- 

t    CO  rl   •rf-  o   o 
in4        « 

^    t^  m  ONOO  00 
tJ-        rl    in  m 

.       O    NO      ON    t^    >« 

°  NO        rl  oo  m 

t^  O   "•   O   m 
•H         to       in 

rl  00   M  oo   0 
M   M   »<   rl 

t^  w    t-^  On  ro 

M   rl   to  t^  to 

rl    0    0    O    w 

00   u-i  t^  n   ro 
X              rl 

i-i  vo    t^  to  On 
rl    t^  rl  00   1^ 

0  NO    irivo    o 

M      ^    ^    ^ 

rl   O   (-~  ro  m 

NO      M    00    NO 

Q    0    0    O    0 

M   m  o  NO   o 
rl              •*  rl 

►.    <7n  rl    M    O 
to  m  rl   w  NO 

0  ^  ^  o  vo 

O^  ^  ^NO  00 
m  rl    M    CO 

M    if  *'^oo    w 

rt*        >^oo    CO 

00  00   in  m  .*■ 

«    «    rj.          ^ 

0   m  0   •+  rj 
rl   ~   T» 

O    t^oo    On  m 
m  t^  )i    to  in 

1 

5     m    u-iNO    0    On 
-.in        .*• 

V   00   •«  NO   «n  0 
•-    ■+  ro  •«    rl 

On  t^  in  Tt-  o 

o    rl    r*^NO  NO    r*^ 

oo   o   0   0   O 

q  mTi-  o  Th 
*  rl  •+  "^  lo 

00   ro  moo   ^ 

rt-   On  r~.00  OO 

rl 

rl   O   0   0   O 
m             to  « 

in  rl  NO    0    On 
-1         rl    ro  t1- 

w  00     to  t^NO 

00    rl    On  On  to 

rt   «   rl   rl 

0   ONOO  oo   0 

t\  "  ■* 

O   On  ""   r~.  m 
rl         M   «   « 

to  rl    M    rooo 
NO    i^  m  i^NO 
■1   rl         »   rl 

O   0   0   mt  in 

rf-                 n    to 

Ti-  m  o   ^  *^ 
CO  cl          4  .*• 

O     O   NO     t^   M 
in  On  —  NO    rl 
CO         c<    cl    CO 

0    O    ro  «A  On 
w    rl    ro  H 

i«*    On  »-  NO    >i-i 

00  OC    ro  f  J    t^ 

On  00    **    u^  rl 

t-  O   0   to  Th 

tJ*oo   rl   to  ir, 
w    to  rl    in  rl 

•4-  O  \D     to  1^    i 
M    M    rl    rl    O 
to  rl   -<   rl 

t: 

t  00  NO   r<   o   *n 
M    H    w          ro 

^     moo    t^  On  tJ- 

t-i         in  M   in 

NO  00  i^no  0 
rl         H   >«   rl 

O    0    "    0  00 

»<       n  * 

00   0   »noo  NO 
to  rl         «   « 

"-«  oo   rooo   m 
0  >-        n  - 
to  ro              M 

m  0   0   O   "n 

11    to         to 

to  ^  On  On  t-- 
to  in        in  ro 

O    i-i    O  NO    t^ 
to  t^  ^  ^  to 

t-~.  m  1^  m  0 

to          CO  n 

t-~  On  m  11    rl 
^  ^  in  to  m 

rl    rl    M  vo  oo 
rl  NO    0  00    to 
to  rl    to        rl 

O    0    0    m*0 

to                    11 

0    -    0    -    - 
tJ-        in  to 

t^  ^  0    O    rl 

t-.NO  NO    t^  « 

rl         n   H 

ro  rl    u->  u-i 

N-  vo  vo    •*•  On 
^  "i-  to  m  ro 

rooo    tl    On  w 
ro  ro  t-^  1^  tl 

m  rl   0    C>  O 
to  rl         to  .^ 

rl    ti    inNO  00 
in  CO  in  to  CO 

rl    O    m  rl    c< 

^  cl    rl  NO    O 

CO  CO  rl    « 

^ 

»  rl   •+  r^  0  00 
n  ^            m 

S    »  00     ON  OnOO 
in  r*N  r*^  m 

*  ON  0    0    r«  w 

00   0   w   0   0 
in 

O   rl   in  M   On 
M        n   >*• 

t-  tooo   >n  w 
w              M   rl 

m  o  NO   o   O 

NO  NO  NO  00  00 

to  11    t4-  to  to 
n   moo  00   O 

M               11 

O    ON  CO  in  O 

in  w 

»l-NO   Tl-  m  t-. 
to  tJ-        rl   ". 

t^  to  On  r-N  o 

n      M      M      M 

0-000 

■>*-00     0  00     CO 

to  tJ-        m  rt 

t}-  tJ-  I^NO  00 

O    r»    0    O  ^ 

M    ro  (^    0    ^ 
U-,  ro  lo  to  -^ 

On  -a-  ^  ro  w 
H    rl 

C>  (--  0    O    0 
rl 

•*  ejN  «   ON  0 
w    rl    CO 

inNO  oo    1^  0 

M                                  M 

«D  00  00  00  >0 
^   O   a,   3   S 

e^i  lo  o  SO  '-o 

00  00  OS  05  OS 
IC  lO  lO  lO  lO 

«D  i>."oo''e-f  to" 

e-)  r-(         1-1  CI 

•  ti  >  !>  _• 

!■-  oo  00  (N  r-( 
O  "-I  i-H  lO  ■» 

CO  XI  50  <0  «D 

Ttl  ■^  ■*  1-1  CO 

■^  lo  00  e^  t^ 
so  «o  o  1^  r- 

CO  50  O  ®  O 

00  t^  •*  CO  00 
.-1  rH  tH  rt 

00  O  (M  CO  'f 

t^  00  00  QO  00 
CO  O  'O  -O  CO 

CO  oTo:  CO  CO 

rH  S<1         i-(  i-c 

CD  Ct  lO  OC  C5 

00  00  05  05  oa 

CO  CO  »  -O  50 

I>  CC  O  rH  Tt* 

rl  i-<  CC  r-t  r-( 

-fcj  b  c  ^  e 

.-1  (M  CO  I^  Ctt 
OOOOn 
t^  t-  1^  1~  t^ 

1^  CO  O  00  l^ 

s>i  i-i  :c      n 
^  c  s  c  c 

C    =    CS    =    S     j 

cc  I-:  ►-5  i-s  >-j  j 

CO  0-.  t-  1-  o»    1 
(?1  Ol  CO  rt  CO 

l--l^l~l-l^ 

^^„^^  _„„__  _„___  _„___  „_„„_  _    ,-_„  _,_„„  ,, 

1 
6 

»H  ®1  M 1^  >9 
19  IIS  »5  lis  »a 

«stN.xes© 

us  iS  19  >9  tt 

»-l«lM.*>9 

«Sr»ar  Si© 
1 

099 


TABLE  XVIII. 

Elemontfl  of  the  Orbits  of  (.'onicts  which  liavc  been  observetl. 


'2 
1 

3 

4-t 

*^         tZ 

J 

4J 

-2 

^    M 

-^    "2 

.5;'2 

d 

%    2  =  -ci 

3     n 
2  -r  >^  'i; 

'3          o,2 

«  =  fe  ^  If 

6   «^s 

-  2  '1 "  y= 

waJwl 

:|       ^  I'- ' 

<i       V       'J                       aJ 

<C            o 

aJ 

i!            oi 

■V        d 

-3       T3       '3                        -e 

-e          -3 

-3 

"3             -3 

"3      rg 

f. 

S      2      2                  2 

2          2 

*^  &-       to 

? 

C3                  '*' 

bi:.j  M^  to    ^  ^           bo 

-*J                 IC 

•w 

00         jJ    bfi  *J 

^ 

2  g  £  s  2    *  S         2 

o  c   y        O 

O    t.    11          t. 

^  g         2 

g.... 

5=000 

C<          o    t-    O' 

Lri     li     L-     O 

-f-i-ffc--rr;           t-ts:-- 

U    ■-     t.    6     -^ 

—   u  —   t.  a 

« 3  K  Q  M       Q         « 

OMQ     M 

Q         W 

Q 

M      QC^a 

«3;:^5 

o 

VO 

M        r^ 

•<»•             to 

00                           ON 

\c  aa  ^   ti  -t    ooooooooo 

OO     ^  tl     tJ-  tl 

0    tl    On  On  to 

O    O    O  NO  NO 

M     n     »^  to  tl 

NO    00      to    0      to 

1-  VO     -<     ONVD 

1-^  in  to  m  to 

«    On  tl    O  OO 

•+*  T^oo   tl  OO 

-1-    ON    to    Ml    NO 

ONr^Ooc-H      sO-Ht^-^r^ 

tl     O     Ov  •*••>■  • 

■o  On  0   OO   00 

ONOO     rt-    tl     M 

NO    o    i-~  ^-NO 

On   -4     t  -  OO     r^ 

to 

ro  r^  cr\^    n        ^\D  00    c^sO 

***  m  r'l  I  •,  <-  ■ 

to  O     ONOO     tl 

tn  to  tl  NO    to 

to   tl     On   tl     tl 

ONVJ     O     tJ-   lo 

c 

00    rl    <-•    Tt-  Tj-      rl    r^  fl    rnsO 

O   OO     O     CV  '  • 

0     O   00     tl     tl 

to  On  lo  iri  »o 

00     -H   00  00  NC 

•I-    »0    to    tl      Cn 

00    ON  1^  t^  m     Cv  ^  "^  t*^  1^ 

O^  O^  O  \D    t~^ 

t^NO     O  00     1^ 

On  On  O    "-  00 

On  I-^OO     on  >. 

00      O    NO      to    lO 

^CTsCT>CnO       C^CT^^^O^ 

C^  iTv  0    ON  CN 

On  On  On  On  On 

On  On  O    O    ^ 

OO    On  ON  On  0 

On  0    On  On  On 

>e 

M 

s 

;i 

o 

00         «   m 

O                     tn 

O    ON  o 

On  0     O     to 

NO                       •.*■ 

M 

0   •+  to 
0    tl  OO 

NO     "4     On  ON 

Tt-                                     to 

NO 

to  to  rh  tl 

^                    to 

to 

1^ 

»  4 

■+   OnnO 

ON  *  tl  OO 

On                        On 

00 

^ 

^ 

NO      ONOO 

O    tl    O    tl 

On                       to 

•+ 

OOOOO       OO0OC\ 

O   0   O    ON  o 

0  OO   ON  r^  O 

O    t^  O    O    O 

On  O    O    0    >o 

O    O    O  00    o 

«--->-■"       mmmmO 

».      M      «      O      - 

w   O    O   0    ■" 

>"      O      HI      •"      "4 

0    ►-    «    1-1    0 

-;  «•  -•  d  -;  1 

t    w  vo   O   «   i^     r^oo   0   0   f< 

On  O    to  O    •-* 

O      y^    Q      m      IT) 

ONOO      M    NO      t^ 

tl    O  NO    •+  ■+ 

H    tl    tJ-  0    to 

M>^i-i^ri-     »^HH         w^ 

tl    ^  to  to  Wl 

M                 n      tl      to 

M      to    tl               tl 

Hi*    N-    »o          to    1 

**» 

V     *^  00    t^^      VO    r^  0    CNVO 

to  tl  OO    »*•  to 

0     w     lo  **■   »^ 

ut  to  tn  O    0 

to    to    to    tl      to 

On  Tj-  1-  NO  00     1 

M    c*^                  tl          un  M    r*^ 

^  to  to  un 

»0          tJ-    to  tl 

M         w    tl    to 

tl                Tl-    -      U-, 

t«4      to    to    Ul 

°  o        ■■l-Tht^oovo-vOM 

On  ■>*■  "1  t)    tl 

O  00    0    "    "i 

M  r^  M   to  tl 

■4-  tl    1-    t~-  .+ 
to  f-oo   tl   -^ 

11    O    t^  to  o 

t^        OO    t-v  wo 

tJ-           ^           to 

M      M    NO    00      to 

to  c^oo    w    to 

t     ult^O\Tt-f>      vOO\OOt-^ 

^r\  Tt-  trioo    to 

O    O    ON  Th  0 

tO06     O     "i-  0 

00    0  oo    to  o 

11    1/1 -O    0     -1 

*^vrjf»ui^     Mrlin        r» 

w    tl          to  to 

ITN            to   to   « 

to    to    tl      TO   m 

M                to   to   tl 

tl     --     CO            to 

^»j-wLot--oo      rtoofiOO 

O    O    to  t<  t1- 

0     «     to  On  tl 

«    lOOO    •*  'I- 

til    M               Tj- 

It    w    O    tl    to 

On  tl     tooo     to 

c; 

cnr^w-rt-M      »r^        Mv/^u^ 

■<1-  tt    to  « 

«     «            Ul  tJ- 

•+           tl  -t 

-i-  -«   to        tl 

wit^^ir^r^     rimT^-oro 

On  OnOO  no    0 

•*  •>1-  ■/-,  1-1  OO 
■<}•  t^  t-  to  o 

r^  t^  «   0   to 

to  tl    to  r^  ly^ 

NO     •+  ■!•  Tt-  to 

000^         ^^     mm-^min 

to    t^    T^    ITN    tl 

•ri  to  tl  00   tl 

tl    tJ-00    r-^  tri 

to  NO  NO    to  On 

1-                     w       rl         H    tl 

w          to  to  .H 

rt             M     n     M 

tl    «   « 

M      IH 

tl                  to    M 

!•«*■•>   t^oo   m     0   0   O   O  00 

<T   ^'^  r*^  l'^              li^  M                    t* 

><J-  tl   0    r-  tl 

ITN  0    On  (-~  * 

tl        tl  tl  ■+ 

NO    to  0    O    l-> 

On  O    to  t^oo 

H- NO    to  0    0 

ui        tl   u-i 

i-     to          N*'  tl 

t/^      tl 

tl    t^  to        to 

^    roi-^x-ic^e)      Ot-~ooooO 

00    to  tl    1-^  ■+ 

to    to    l-«    NO      tl 

tooo   r^  t^  ■>!• 

l/N  tl     1-1     to  to 

"l-   -4     ONOO  00 
*!-  tr,  tl    to  to 

13 

r^w-iMH.                    Tj-ir^t^M 

to                    U-l   "1 

M      «      M      «      tl 

«   tl   « 

to    ti-,    M 

r^  r»   t^  r--  r^     i^oo   ri   t^  m 

to  ^  ^  tJ-  uo 

to  NN     t*-nO  OO 
^  ty-i  'tr  ^O  o 

•+  0    to  I--  t^ 

NO  NO    OnnO    O 

O    On  I~~nO  00 

O'-'OntJ-oi-^     Mi^r»v£30 

wi  to  o  00    <-i 

0   «   r^  -  00 

>i-  t)-  to  ji   to 

OO     O     On  to  li-l 

t«         tl   w   M      tl   t<   -4   tl   m 

w    ti^ 

1-t     tl     11     to  tl 

M      H4                 to 

tl    tl    tl 

►.     tl     w     11 

•»  tJ-\0    t^ONON      ^w^CTs^ 

00     M     tl  00     vo 

0    to  M  NO    O 

On  O    O    «    0 

to  O    On  t^  O 

NO    to  «    M    0     ' 

iHi-iwr*^iH       tl          r*^r^^tn 

LO    Ln   N^    U-l    1-4 

m  tl    to 

1-    r^  i/N  tl    tl 

to           ir^  tl     li-l 

tl    rh  >o  to  »« 

g   «    O    ►«    i"  1-^      "100    »1-  t^  ■* 

to  00     m     tJ-   tl 

M     to  tOOO     On 

vO    to  M    to  T^ 

to  N*    w    to  o 

f-oo  OO   t-~  r^ 

ttttt-iu-iui      H«^>^«i-t 

to  ^      U-)  T^ 

^  ^  to  to  to 

Tl-         to 

M    tl    to  to  to 

-1-  4^  to  to  to  ' 

■C'^O'+t^O     CT\>"I^fnm 

0    H  00    O    to 

00    to  ^  O    to 

iri  ly^  ^  Ov  tl 

tl    O    Tl-  M    " 

t}-  t---oo   o   to 

h 

tl    «                  M    t>               m 

tl  « 

tl    w 

tl       tl                 «4      >H 

tl  -1 

OOOOrtCO      COODr-rHlM 

t^  CO  oo"^  iM 

t^  CO  t^  t  5-1 

05  CO  U5  lO  '1' 

O  00  t^  OS  CS 

-1  I--  i»  O  CO    1 

rH  (M                     Ul  i-H  (M  rH  1-H 

W  rH  IM          .-H 

--1  M      1-1  ffl 

t-H  rH          1— t 

C<5  5-1         51  rH 

5-1  5Nl         CO 

.j:.^    _    ■          -d 

, 

l-H 

1 , 

o  a  J:  o  "3 

-O  "S  ^   M  > 
U    0*0    s    O 

fc-  *^  o  -^  1^^ 

■g  -a  -g,  M  c 

-^^4  a: -*!'-! 

£<  o  -=  O  O 

c£  ;^  1-5  fe;  t^ 

ClCCCO'l't^     OOQOI^OCOJ 

05  C5  C-l  CO  -t< 

C0  50C500 

i-H  oi  eo  "*  C5 

O  O  I--  r-  J(5 

-f  l.'^  lo  CO  CO  ! 

f-f-ti-fM*    -ti-fioiraic 

lO  lO  tr  --D  -o 

CO  CO  O  1^  1^ 

t-  1^  i^  i~  i^ 

CO  ;»  cc^  CO  cx) 

OC  CO  CC  CC  CO 

I^I-l^l-l-    1^  t^  [^  t^  t^ 

1,  1^  I-  i^  1^ 

1-  l^  t^  I-  t^ 

(^  i^  t^  t^  i- 

t^  t^  t~  w  t^ 

',:;  i:; !:;  !^  S  1 

iHoios^iis  tfi>>ces%o 

tHINOS-"*** 

«0  (N.  X  s»  © 

1-1 91  as  ■<*  1* 

«I>.XOt© 

-H  (51  M  -^  »« 

O 

t'«l>»t»l'>I>>    t«t^l»l»(X> 

CC<X)(X>(X<X) 

ocacccccA 

Cd  0&  C^dd  c& 

0&Ae»eio 

oeco© 

2 

IH 

1-H^^rH  iMi^ 

640 


TABLE  XVIIl. 

Elements  of  the  Orbits  of  Comets  wliich  have  been  observed. 


a 

.  0.2 

WM 

^ 

tr. 

w 

O 

tH 

U 

a 

»-^ 

a 

vt 

r^ 

o 

r*-i 

<^ 

f-i 

M 

VO 

oc 

t^ 

NVl 

O 

•+ 

1/^ 

■   v/- 

r' 

rl 

O 

0 

vO 

vri 

»n 

>  o 

O 

^ 

<J^ 

o  o  OO  o 
M  M  d  »-* 


»    rt   '*■  o    f^ 


»i^    t«^  m  tn  I 


^    O    t^  r*^  O 


wi^O    O 


^    ^  ■+  '^^  «^ 


+  ^0    r^  O    O 
ft    in  r*^         ti 

_<     CMM   OO 

D     On  r-^  00 
O    CN  ^'^  '^'^ 

M      rl      M      M 


O     to  —     M     O 

•^  tJ-  in  <j-i  c*^ 
tJ"  f-oo    O    ro 


1^  i»  O  CO 


ll^^l 1 

-r  1-  lO  tt  "-3 
K  cc  cc  cc  X)   , 

seeeo 

FH  T-<  Tl  FH  TH 

i 

3 

a 
6 

Zach. 
Piazzi. 
Saron. 
ly  Arrest 

1 

-3^  a  a  £i 

c£               3 

WO         W 

a  H  <  ^'t  w 

■J 

mil 

s 

% 

1.  Jo. 

1    1 

|a|| 

3J          <U 

1  1 

sir 

6 

5«S  ^ 

,5i 

hi 

1    S^^ 

w 
5 

't-oo  >o  »o  o 

Wi  0  OO  00    o 

6\  6  6\  6^  6 

M  so    0  so   ? 
00    »rs  u^  ro  t^ 
OS  T»-  -o  t-,vO 
M    M    lr^  ITS  s^ 
O    "«  00    0    r^ 
OS  M    OssO    M 

OS  o'    cfs  0>  o' 

so 

"h  -    OS  O    OS 
0   "soo   r-^  ti 

to    NN      Th     fOOO 

^00    «    vrs  M 
tl     OS  f»  OO     OS 
1^  M     t^  SD  00 

OS  0    OS  OS  Os 

t^   to  '•f  ^  OO 

tooo    0  so    in 
•i-  "*-oo   o  00 

•t-  in  1^  Os  OS 

tl    Os  M    ro  tl 
Os  r-.  Tt-  q  q 
OS  Os  Os  o'  o' 

OO  o       00 

so    t1-   Os  in 

-      TJ-OO      11      M 

O   vO     .*    to  CTs 

tl    I^  "t-  0    o 

lO  in  fo  M    CTs 
in  Os  0  00    m 
Os  Os  o"   (7s  Os 

OO    Os  m 
O    m  t~-  in  CTs 

1-^  in  M    to  o 
OO    to  M    rl    rt- 

to  Os  in  Os  O 

00  OO    M    Os  OS 

'T  ?  '^.  ".  '*'. 

OS  OS  0    0    OS 

OS  rt    OS 
r--  M    o    On  O 
m  tl   M  so  so    1 
m  csoo    f-  tl 
T^  Tt-  ro  in  in 
•+00  00  OO  so 
OO    O    O  so  OO 
Os  C               .     Os 

« 

0    0    0    0    0 

M 
IN 

to 
0    0    O    0    OS 

00 
rt 

00 
00 
00 

00   0   0   0   O 

O    0    0    O    0 

OS                      M 
tl  00          00 

invo   tl   r^ 

t^   O  00  00 
M     t-    -1     Tt- 

so   m  0   in 
Tt-  Tt-  ►.   Os 
CO   r^  O   OS  o 

O  OO 
tooo    tl 
to  o    11 

OSM     -t- 

O   (^  m 
m  tl   Tt- 

Osoo    m 
0    O    Os  Os  OS 

OO 
so 
Os 

M 

H 

M 

0    0    CA  O    0 

m    m    p^    m    0 

0      M      M     M     M 

O    O     M    o     M 

M     M      0      O      O 

M     M     O      M     M 

*«* 

i    »   c  Th  i-~  M 

^     w  t^  0    tJ-OO 

M      p)      C^    tJ.    U-l 

n    00    H    ■+   OVO 

t-N  t~-  ■>*•  0    0 

M     Tt-    «               M 

u-t  Ti-         tl    ro 

ro  OS  OS  0    w 
vO    ro  ^so    sn 

0      to   T*.    t)     * 

to  to  to  ^ 

rt   Th  o   Tf*^ 
■*  un  Tj-  •+  M 

to  ^  O    to  r» 
«  so    in  ■+  rh 

r^  Tt-  0  r-  o 

tl                        Tt-    Tt- 

tl   mo   O  00 
tl         tl 

M     t^    M     t^Si) 

in  r^  tl   in  m 

O  Tt-  tooo   r^ 
to  to  to  tl 

toso    tl    0    to 

to    to            M     Tt- 

M    m    toso    ^ 

Cs  m  M   M   to 
in  M    tl    M 

00    M    N    l^  1-^ 

MM             M     m 

OS   M      to   M      tl 
toso     t^   to   t^ 

to  tl    Ir    o    T^ 

to  M    m,  tl    tl 

to  tl    OS  m  tl 
M           tl 

~     M      Tt-    to    O 

tl  00    t1-  ^  tl 

a 

V     i-  \o    ■+  O  00 

vo  ^    f)    rl    t^ 
0    O    i'^  m  f^vo 

H    OS  ^  0    H 

-    H    •+  OS  0 

ro  o    rooo    rt 
ro  osoo    0 

w  n  •« 

H    so     t^    M     O 

tl   «   to  rt   to 

OS  «  m  rt   0 

to           w     n     to 

■>*•  r~.  OS  M    OS 
to  «   tl   H   Ti- 
to         to  W    t< 

tooo   0   osoo 
M        to  m 

to  o  OO   tn  t-~ 
t^   to  tl   ►«  Tt- 

OsSO    Tt-  0  so 
Os  rt  ■♦  M  r- 

tO           to  M 

O    OssO    M  so 

M      M      M     M      to 
O  SO     to  t^oo 

tl   •«   tl   Tl-  m 

Tt-  >i   tl  so   tl 
to  m  f  1  so   N 
to  H   to  tl   to 

m  tl   Tt-  tl   M 
M         tJ-  in 

M      M      Tf    M      M 

M     H     tl 

t1-  o    0    to  to 
tl    M    Tt-  Os  m 

to    M                 tl 

Tl-  m  Tt-so    O 
tl    M    to  in 

00    0  00     Tl-  Tl- 

■+    Tt-    tl       M 

0    tl    to  to  O 

so  Tt-00    tl   m 

to  tl 

h 

5    OS  !-»  'I-  "1  r- 
u^  .!i-  ro 

V    "J-oo   cys  ^  ■* 
r~.  On  n  00  w 

0           OS  P*    *n  w 

M 

t^  rt   wt  0    c^ 
rt    ro  CO 

to  o    n    H   * 

Th    t<      lO  •<}■    ITS 

toso   tnoo   M 
r-  to  to  ti   t-^ 

O   tooo  r~  r^ 
tl   M        in  f* 

•-    ■+  t^so    I-. 
"l-  •*  tl        tl 

so   tl   OS  in  ^ 

m  OS  -^  O    to 

mm           m 

so    so     O     Tl-    M 

m   -f             m 
00    tl    OS  Os  Tt- 

tO    tl     Tt-             ^ 

to  0    to  tl  00 

Qsoo     ro  Tl- 
M     M      to   M 

T*-  m  Th  tl   t-~ 
tl    tl    tl    Tt-  m 

t^oo    to  Tt-  tl 
Tt-  N         m  M 

so  Os  r^  o  Os 

in  o    OS  r-.so 

MM                 tl 

O      tl      Tt-     t^    T*- 

m  Tt-  to  tl    tJ- 
00    Tt-  o    1-^00 

to  tJ-            tl     m 

tl    H    m  1^  tl 
in  in  t-^  Tt-  OS 

00  00  so    to  o 
in  to 

so  to  M  m  r^ 
m  «^       to 

Os  t^  Os  r^  m 

so     OS  Tl-sO     OS 

1      M      tl 

Eh 

•9  OssO    0»  OS  OS 
g  OO    U-l  I'^OO  so 

■<    OS  t^  t-~00    t^ 

m                 M 

»!•  u-i  OS  O   11 

so    O    1^  H  so 
tJ-  m  tJ-  w 

u^  t«    t-~  0    •'I 

m          M 

M   ri   0  SO   to 

m  t-v  >-  00   t^ 
to  ^  to  in  N« 

0   Os  tl   «   to 

mm           MM 

SO  SO   0   »n  in 
to  Tt-             m 

O    to  to  toso 
in       tl   H 

moo  to  M  •^ 

M     M     tl     M 

o  0  n  Os  Tt- 
in  M         m 

Os   M      (JS   to   tl 

m             Tt-  m 
M    t^    tl    t~  tl 

M     tl     tl     M     d 

00    M    H     t^   M 

in  to  to  M   to 

O    to  o  so    M 
tl    M    Tl-  ro 

Tt-   tl   so     to  t^ 

tl 

O    t^  tl    0    0 

M                t1- 

OO   M  OO  OO   m 
to        ^  M    in 

tl   0   tooo   0 

M      M      tl                 M 

00050  00 
rH  l-H  (M  r-l  C<l 

1^  =0  CO  O  O 

cr>  X  cTj  cnci 

O  5-1  (M  CO  CO 

O    Od   O   Cd   O) 

1^  1^  t-  t^  t- 

lO  to  t^  CO  00 
Ci  C".  05  C5  ov 

I-  l^  I^  t^  t^ 

t-Tisoo  oseo 

OS  05  --I  5-1  Tf 
OOiOOO 

t-..  i~  CO  00  cc 

f— t  i-H  r-i  rH  r-! 

i-i  1— 1  00  00  e-i 

52;nftMS 

lO  --0  tt  1^  00 

O  O  o  o  o 
CO  cc  CO  CO  OO 

1-1  1— 1  1— 1  r-^  f-H 

IM  CT5  C-l  O  lO 

tH  (M  ?— 1  rl  1-1 

CO  O  ^  n  ?! 

O  rt  -H  n  ^ 
as  CO  00  X  -JO 
»-1  1— 1  1-1  r1  1— I 

Tt  05  10  rH  t^ 
1-1  ?l 

cc  ^?  >c  -xj  CO   : 

X  3  X  00  X 

1 

©©e©»^ 

rH  1-1 1-^  1-(  1-H 

« t^  «  a*  © 

1^  fH  ^a  tH  91 

r-  ©ISO  ^  »e 
(51  (N  ©1  IN  91 

«Dt>.X©© 
91 91 91 91 «« 

^  91  M  ■«*<  »S 

CC  €0  w  w  w 

C550  W  ^  "^    , 

41 


641 


TABLE  XVIII. 

Elt'tnentrt  of  the  Orbits  of  C'ometn  whieli  have  been  observed, 


\ 


i" 

• 

.a 

C 

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tl    TO  n    w    M 

w    in        rj- 

tn  m  »!-        m 

•COO   rl   0   0   m 

w   m  o  n   ON 

M    ro  M  00  so 

m  0  NO   m  rl 

rl  oo    CO  On  in 

«-     «   NO     0N05 

NO    to  On  fN    tl 

E^ 

M         M   rl 

MX         r< 

M               M 

M      M 

n        ^   ^ 

>«   rl 

eOUiOi'^  N 

<M«OiMiC  t^ 

CD  t^  C^  I^  CO 

00  1-1  ic  cr(M 

O  »0  lO  1^  rH 

lo  oi  ©  c^  (sT 

C^Ci  '^  X  --o    ■ 

Ca  rH  rH         i-H 

rH  rl  rH  iri 

r-i          r-1  rH 

CI               C) 

i-iOl         (M 

CI  CO  r-t 

»-"        0}    . 

J 

^ 

J 

bb  >  u  e  h 

0.0    S-S    O 

1-5 -»!  Hi  ■<  1-5 

lO  iC  oo  o  © 

©  ©  N  M  CO 

CO  CO  -t  -f  'Jf 

lO  iO  iO  lO  CO 

«o  ;o  «c  ©  o 

©  ©  r^  t^  t^ 

t-  t-  1  -  CO  CO 

Tj<  -)(  -t<  •^  -*< 

•!<'♦<•*-»<-»< 

-11  -t<  -f  I*!  -f 

*tl  -f  -*  ■'t'  -1" 

•^  -»<  f  -*<  -fi 

-tl  -f  -t  -r  -t* 

CC  00  00  X  CO 

00  C»  CO  OO  00 

CC  re  OD  CC  ZC 

CC  CC  CC  CC  :c 

CC  CC  cC'  CC  CC 

CC  CC  CC  CC  CC 

00  CO  X  X  CO 

,— ,,-^,-,^^^,-,^r-,.-^T^r-,^^^^             rr                      .                   ^                                                                                ,j 

»r>.x«t© 

1-4  (51 M  ^ »« 

«eiN-GC«tO 

1-1  (51 M 1*  »(5 

»r>.xet© 

-Hi5lOS'«i(S 

(»r-X©©   1 

o 

r>  ■>»  [<•!>•  QfD 

(X  x  an  OD  (K 

00  X  X  X  «S 

cscsct««t 

ctoseseto 

©«©©© 
(51  (51  91  ei  (51 

©©©©-- 
(51  (51  (51  (51 81 

643 


TABLE  XVIII. 

Element  of  tlie  Orl)its  of  Comets  whicli  have  been  observetl. 


>> 

1   1 

a. 

e 

S 

.         § 

• 

3 

5 

d 

•v   ^   S   r   :  - 

^,  yi  p.,  ^  ^ 

ilisl  ^<:^1^  O-^"^-^ 
BKi^al  c2-2;2s;5  H^'^Sx'W 

S  ■"  "  .•    »^  £  K  ■»■:„• 

(jj            q3              <v                           a5 
-O          '^             ^                          •§ 

T3             -3          13 

^                               s^                               g 

c3              eS           rt 

u              u           u 

1    '^ 

^ 

St  ^J         CO  ^      ■*■'    ^C'  >.    ^    ^      -*^         00       ■*-! 

iCU         bC^      SO  _  *J 

u 

CC^           0=:        >vO=^=^5:        t)           0=:u 

Co         0   u      0   ==    Sj 

i  ^- 

jU    _____    __ 

Co'-"                OtH                               OJ-c^ 

ijS^ijS     "^       "^^^ 

e^a     f«       pp:5               Q     W     Ci 

«Q  ""wo  ;!    3 

O    O           O    Ov 

-«    0    r»    r*-  ^ 

0    ^  rr.  5    rl       Tt-  e*-;^    «    ^            w,    i-v         O 

0      M      M      t->         U1               f1               M 

v^  rl    tr-i^xi    tJ- 

00   r--  r-oo   ^^ 

•4-  H  vD    r*    i-^       i^vO    <:#-  u-  T^      rh  »/*.  i-l    O    O 

00  0   o^  r^o     00   0  ;-•  0  00 

(..  -    rt    0    r-. 

ri    H    r*   r-  o 

0    «     ii^OC    *^        f^  m  rl    •-'^  ft      00    >/^  rf  t---vo 

t^    t~-  0    t^  fi       rl  00    f^  rj-  r» 

Oh 

0    ">    -n  o    M 

^  ^  «    _    r. 

■rh  OC  OC     «     rj-       «     r»   y;     1  /-.  -H         m  I--.   «.     C^  t-- 

r-^OC    DC     r*     fl         "^-OO     fO   OnOO 

to 

tl    •*   -    rr  M 

O    O-'  m  ri  oo 

0    'i-'^  ry.  on       t-  I--.  0    i-i    —       Pi    fi    •-<    !-•    t^ 

0    ('-  ri    "^  f*^      0    r*iDC    0    ^'^    ' 

o 

00  SO    "->  c^  »r, 

""  ^     C7\  i/~i  f) 

■>D    »''.  c^  t'l  »/■■    00    '■"  "-^    rj-  "^       0    t*".  •*■  'n  H 

c^^o    (>vc    t--     v^  0  sT)    m  oc    l 

•—1 

CN  O     ON  O    1- 

t»    O    t>  «    »J^ 

0  0  0    0    Ca      Tl-   ri    ri  rt-  crs       as  «    f'^  r--  »n 

q  00   t--  ».i^oo      r-  q  q  q  00    \ 

r^  d  a\  d  c^ 

d    O'    C^  C>.  O^ 

ds  dv  d  d  ds    <:>  (^  6  ds  '^    (^  6  6  c^  <> 

6  c\  6s  <>  6\    6s  d  d  d  d^ 

m        <^ 

^    T»-^0         vo 

rt-'O     rt-  -+•       »/100     -n           "O      i 

»C    0    - 

O    i^-  w:           ri 

^  1/-  ^c    el    0     ■^-  X)                        rh  ■^  "-r  r-^ 

1/1  ^  '^  00     >■■<        r^  M     CN  n  rO    j 

00    r^.  lO 

vo  'sD    v^        r-. 

0  \C     I-'     *-     'H         Or!                               r)     0    3C     C7S00 

i-r^  -«   ■>  cTi  t-^-     »-   c^  0   0   0 

V 

DO    00   OO 

O  cl  >r:         'O 

»>-.>=crs-^c^^rl                          m-fJ-r-CNt^ 

r»     rt     M     0  r*^       0  OOC     0  M      1 

r-.  r--Dc 

-+■  O^'           1- 

rl     V/IDO     0     ^       (^  »-<                               c^'C     i-n  ',--:  t-. 

r--   C7N  ri  00    0      ^O  ".D    0  0    rj- 

CJv   J\  ON 

iri  m  Cn          tJ- 

m  ui  —    0>oo       rl    0                        Ooc  -o    0    •+■ 

0  Os  0     0^00         Cn  O-  '^1  tl     ^n 

o  o  o  o  o 

1*1  v^     O  0  oo 

0   ^-  c^'  CTN  ON     i--  0   0   0   0      0  a>  it^  c^oc 

CN  000    c>  On     OS  C^^  oc    r-.   ; 

"  d  c  d  «■ 

o  d  d  -■  d 

«  d  d  d  d     d  «  «'  -.'  w     d  d  d  d  d 

d  d  d  d  d     d  d  d  d'  d 

s    mo    Ov  -*-  CI 

C<  OO    f1    t-  «-o 

CTC\00f"1       wrf^r^N-        0000N''-*'O 

C>  r'l  m  M    *J*      M    1/-.  M    0    ^'^ 

M      U-^    W      f.      ir^ 

rr\                   (-*-,  (Ai 

M           cl               »-<    ■'j*  ■^  "^  Cl        c^  »n  t-(    vr 

>^    M    vo  ui  c(       rt    '-n        M    ui 

'" 

..     f»   t^  Ln  >H  00 

►^    'O  c^.oo    t^ 

M    ,n  i.^  r^  C7S      0    OS  0    r»  00       «!t-cc    -i-  ^00 

MVDoc    r^vO       rooovC    rj-r-- 

ir,  M 

r*    \r\          in 

^mir.  N<J-rou^         fi            ui         t) 

MuirJ-inri-            TJ-vor(Ti- 

.,    in  r--o  CIO   0 

M     I'^CC     c*^   rO 

CsHOOl^      «0*^fl"*       O^^tnro 

0    t--.  000    M      VC    !>.  r^  -^  0 

•'    OC  vo  VO  kO    ^ 

M    I-*     ro  1--   « 

'^w^'Svi»o^'sOoor^     ^«vnrtw 

McoNu^f*^     mc^^MioM 

s    ""  r-.  o  CO  M 

\D    ^*    m  0    « 

M    hs  \rioo  \0     oc    rt    t^  fo  o> 

vn  f*          f<    fi 

►H    1/1  r<i  t'^  t» 

mroMtlM       «1-Tt*         »-(U-j 

^     f-1    r*^  rt    f  1  Cn 

•-    -(J-  O    *•'     t-') 

CN«-tOt*it-^     MinO    r^cc     OCT^riOO 

■«*■  0  >n  ►•.    (;>     i-~-  00    r^.  m  p     [ 

c? 

i-t    rn  m  i4-i  »y-i 

n  rt    Tj-  rt    f» 

rl    ui  PH    f.  m     c^              rl    •;*•     rl         ■«♦••-.    rl 

t*i        '^■'i'^"     «nMfi          f^t 

^n  fl    O    f»    »'•) 

a^OC    r*!  -f  -*- 

r^i-n^jDCso      oor--inr-.     rf-oooso-^ 

M    r*i  M    m  0       ^  Ovoo    Cs  f*"- 

0    "<   O    r^  ir>  0 

i-»Tj-';fsD'*-       '^   tl     ti     ^     ^       ti     <^0C  ^     m 

•^♦■^OrtO        mitO'^vOm 

n   ti             M 

r)    Id    cl          rn 

CI  N    to                 »-t    t*    ti    t*i  (■n     m  el    w    c»    tn 

(^    M                 rt                      M      M      Cl      M 

1 

;;   vo    vnoo    in  I^ 

O    iJ->  OO    tl 

CO  vO   fi   c>  r-     o^  »■'>  c^   ri-^o     f»o   >-n  m-*.o  n 

r<i>tOMM      OoOO^/^Tt- 

v^- 

•^     W,   Tt     T) 

»^-H-<j-i-im     i/iunu-»—             Mti"^'<i-M 

Mi/irt          tn      r^M          i-iLTj 

^     'f  »^^0    u^sC 

rl    »/^or;  »C    M 

fimvovo^C       t^r^r*^MON 

I: 

»-.•+•        rt   w 

■^^  »r  u-i  ^  u^ 

-•t*    -^     t-1    ^                    M                 W)     W^ 

f-i  li".  f ..  fn  (;> 

c^  r»    O  oo   t^ 

fjo    CTi  r<i  m  »H       0    rl  sD    t«*.  rl       ^  t/rX*    0^  t^ 

00    t-  «    tJ-  rt        lil  tJ-  M    M    w 

,j   so    (-ns^    i^c^ 

^  f  I     M    w  u-i 

^■0'***^'0       MO'^^wr-.      0\^    '      -^lo 

t»    O    t) 

f*-,  r^  ro  1- 

t»w         ►^H'^t^        c<r»            iH<jMi 

M    rt            rt         ro  M    rt 

tfl   C-sO    v/^i^    O 

1^  u^  r*  00    *^ 

ooooC^OO     viM-t-ort 

r<i  rl    w    M    M 

M         M         \/^     1/1     ^^ 

10  rn  M    M    xi-i           rn  M    rt    «.*% 

£  "*    r-  f*^  0    -^ 

in  CTs  r-ftc  ^ 

inooO^C^     r*--CN3«       rowooocO 

OOrOrhm^t-      r-.  rtvCTj-^l 

(-n  rn  1/1  t}-  i-t 

H    f  \^ 

Mr<i          irir*iirir^i-i(^              mm           'j^^ 

<•        t*i  wi          ■^i*  t^  m  rt    ' 

<  oc    Pi    ^  tl  c-e 

ft    fi     i/>  0     OS 

inHOOt^ON'^'^t-.Orl        oli-«OTfr 

Ooe  vOr^iro      MMOrtw 

•-           M 

n             "^   « 

M      (4      M      f1      M            M      M      .it                                 MM                 t-l 

M    rt    rt       rt         »H    ► 

•tr.Kr.K.              -.^.r..       •«•,             •^r^•>.■kK 

tx 

oT-c^oo  cc'oT 

T-.rcr?co"'t" 

Ci  (M  S^l  CO  Oi     -H  CCM  T  'M      r^  i(:>  LO  Oi  ^H 

if^rr-Too'i-Tco'  o'cToo'co'Tr 

»-  TI         (>3  i-H 

<;■»  cc>  — t 

—  Tl  rH  Tl                     .-.          CJ  Tl      (M  rH          CI 

CM  Tl  Tl  t-H  <N     CO  »-H  T^  T4 

^  !"] 

r^ 

^-d 

•-J  ''^  ^  '-7  0 

*-  ^  to  i:^    H 

^r!^f2;l  c^e^l^  a^pSl^ 

'~.  C-J  0*5  o  o 

r-*  T-^  r— <  T—  Cl 

(M  71  iri  cT  Tt    00  M  ~i"  '•<  Tt<    »«<  "(<  10  1."?  ira 

10  I-.  t^  t^  I--    t-  t~  1  -  ce  CO 

f  -r  -t<  1.0  lO 

»C  »0  »■?.  iO  »c 

irj  ic  ic  !:■?  iC     lO  1"?  il'  0  10     0  10  lO  10  10 

00  cr  z-  cc  00    a:  cc  -Xi  00  ■»    oo  gc  oo  oc  oo 

10  »0  »C  10  lO     >0  i^  IT  >^  »0 

1 

cc  X  <x.  X  ac 

QO  X'  'X'  00  00 

ococcc'ooao    oo=oxx» 

-H    I-.   1— 1  .-1   —. 

^ ][l^ ^^ 

1        —'JlMfi* 

•^i^T.caS 

^sier'H«>'?    ci'»/i'i*   ■r^slM'H«>r^ 

«i>.arrtO    "-siM-t*!* 

i      o 

'             1.1N  f"^  1«-1  *««  ?-* 

_<  _m  iM  f*.] 

SI  SI  CI  SI  SI    SlSlSl'flCS    «f"^WW 

«*5!<5M'*    -ti-^-^-i.'^    i 

1     ''" 

1      rjj »» ^1  (Jj  <« 

SI  SI  51*1 '51 

SI  SI  SI  SI  SI    SI  SI  01  SI  «1     SI  SI  1*1  SI  SI 

SI  «J  SI  SI  SI    SI  SI  SI  SI  SI    i 

1 

1 

- 

r 


o 


cs 


1 

1 

1 

.1 

6U 


TABLE  XVIII. 

Elements  of  the  Orbits  of  Comets  whicli  li<ave  been  observed. 


tK     CJ     ■!.  _,* 


<'M^^ 


0  ^    *J 


3  OS  ;--  O  (» 
)  oo    r^  ■*  rl 

4-OC  '^  CnOO 
3  C^  DC  O  *^ 
j^  O   VO     -  OC 

r-  o  q  q  oo 
js  O   d  o   c> 


/^OO 


1       ^ 

C^  r*^  f^ 

w    O  O  O    O 

O  c>OC  O  •- 

a\  o  »^'  f»   *'"' 

OS  Cv^  30     t-- 

d  d  d  d  d 


•*■  0 
ti-i  11 

oe   H 

t-.  fl  ^ 

I-.  ro  C     1 

rl          to  1 

M     n  '^^  ^     I-' 
It     •-     tl     « 



OO  ^ 
CO   P-1 

0\  »r>  ■<}■ 

t-s      to 

ro  M    Ov 

K/-)     -O 

o   ■± 

r*>  1/^  *n 

TO   .-<      C* 

rh  0    r« 

».    M    >o 

l-o  ro  H 

M      h>< 

Ov  rl    w 

M 

M      »' 

o'croo'co'cr 

t-  t^  I  -  CC  CO 

lO  ifti  1"  ''    i'^ 

X  oo  t:  t:  » 


1- 91  «•♦<>'? 

^  •*  -t  -*  •*! 


-3 

?"' 

•3 

i    -5 

1 

^ 

s^ 

^     ■         u 

11 

v 

^    1 

c 

.  c      «-; 

.—     .              at 

>^»  —         ^ 

1 

O 

p^.ij 

.  ^      to 

■11. i 

•I  i.1  .1 

~i?  S^     ! 

Ef^SwS 

^   C   X  i^  W 

X  Z^     ^ 

^  PH>H 

'5^-/^  J 

Q 

6 

<w            aj 

C;                      OJ 

o 

3! 

tJ 

'^ 

-O          'g 

■T3                     "C 

TS 

"3 

1     ^ 

.  s 

g 

£            2 

2 

■*J       to    ^       4-J 

to      ^J 

to  jj        fco*J 

SC.           -h!    M 

*J                tc 

^  "■  -t  ^ 

*J 

o  ©a    o 

2*  g 

o  o        o  o 

O    5:    S    O    O 

CJ                 o 

5     t     0     0    5 

S:     0                         i 

s 

0)  C        o 

Co        »"  K 

u              O   t- 

<H                              Ui 

0  ti 

0                1 

t<  -t-l        I.  a 

TTl         *H   zi    t 

f  t-  t  f  t- 

•"                t-  V 

t"  s   t  a  ■« 

H    w 

»-      i       5; 

•J  3;       -^ 

•V        •-? 

0*    .2                    il    .-4 

O              ,—    O 

.  —  ^   ^  ^   o 

.-.     D 

o«    o 

M     « 

KQ      MG 

«      an? 

ft             M 

ft;^; 

ft 

O    H    rh  •+  OV 

m                  M 

HI     O           iO 

oo             ON   1-^ 

in 

r*-i  t^         rl    M 

CO  h^  On 

^    t^  O    to  ro 

.^  lO  C\vO     O 

oo    •+  co>0 

00    ^  in  Tt-\0 

NO    -<    T»-  Th  rl 
lO    I-.  •+  o  NO 

0    0    cl    ir^  r- 

M    0  NO    0 

r-  0  oc    on 

CI  '^    -.    O    t^ 

-    r-  .-   th 

n  *iJ    [^   CO  o 

0     't'OO     f*^   0 

00    t^  00    0 

C.1 

VO   >3     fl     ^    "H 

t4-  rl    ti    t^  in 

»n  n    o  1X3    to 

>J     -f  ■*   cl   lO 

H.     to  •+   O    00 

cl      H,      CO    CO    W 

r--   0     C^  "^    r» 

NO  OC     in  On 

to 

rl    »r>  rl    r»    cn 

Cl  "^   in  ro  n 

00    Oi  t-.oo    rl 

00    CNvO    t-  'r) 

On  invO     0 

o 

oo    r^vD    ro  rl 

vn  o    1--  HI  ^ 

l^li?      —      fl      CO 

ONOO    O    O    rl 

c^  -f  oo    n    rl 

u-ivo  oc    *1-  « 

00    rl    CN  in 

*" 

O     t-     ^^   »^  H 

HI    <o  o    -    'l- 

^  ON  J\  CA  in 

C^  ON  O  On  o 

r-.o<5  0=   -  00 

cx  c>30   q  '♦• 

On  Cl  -.  q 

d  6\  d\  6\Q 

d  <>  d  d  <> 

CTl  C^  CTi  CTi  C^ 

(ji  oi  c>  cfi  d 

On  On  Oi  0    OS 

Jx  C?v  CTn  d  00 

On  d  d  d 

\o 

Cl 

•i- 

1 

fl      LTi    0 

«    f<          •*■ 

oo          O 

Oi 

M      fl 

00    rl    HI                 I 

' 

CO  N*  \D 

0 

CO   CO            c\ 

O          t-. 

ON 

t-^    r<^ 

ONOO    in 

<7v  O  <o 

•* 

lO  00          o 

t-      * 

•*■ 

r-~  v^ 

M    to  in                1 

u 

c>    CO  h. 

cl 

■4-  CO         t-. 

rl          ON 

NO 

t--  0 

.«■  m  0 

vo  vo   r-~ 

t-. 

CO  in       \o 

-           On 

0 

^   0 

m  r-   ON 

O  »*•  <o 

o. 

oo  00         ■+ 

nC           Oi 

0 

OS  C7N 

0    -n  T»- 

O    0    ONOO    in 

0    0    0    0    o 

0    O  CN  0  oo 

O     ON  0     On  0 

q  q  q  q  q 

CN  3^  0    0    0 

ON  moo    0           1 

>"■  «■  d  d  d 

-  -■  i'  >-■  d 

«■  d  d  -■  d 

„■  d  -.'  d  "' 

0   d  -'  -«■  '^' 

d  d  d  -■ 

t      0^  rt     CTv  *^  w 

r-.  in  u-i  Thoo 

O    HI  00    CO  0 

lO     to   tOlO     CO 

•*    «    10      t~-    ON 

^  rl    ^  ►■•    « 

0    M   0    <^  0 

in  r^  in  0 

' 

tJ-    Tj-    ^    «      M 

CO  ^  in         CO 

CO  cl    fl 

f  1    to  in  in  .^ 

-^         ti    rl    r* 

to 

1  ■- 

^     O  K    M    rh  c) 

\0    M    »n  CO  i^ 

^  mic   r--  in 

<1-  m  rl    1    rl 

00    inoo    On  1 

tl  00    r^    t--  fl 

00    rl    H    t^ 

to                 rt 

M   ^  cl    in 

in  H    H    rl    rl 

rl          rl    HI 

tr^  M    vn        t<\ 

M      rt      M 

1 

-      rt     0     CO   CO  M 
"     C<  00  VO     HI     HI 

•H      CO    a\f>Q      CN 

00    Oi  in  N-    CO 

r^  'O    rl    in  t--- 

inoo    ^  CO  in 

^   0  00    t^  r-N 

1^    M    00    NO 

1 

1 

«  00   r^  rt-  t- 

cl   r--oo  **•  « 

VO     ^OO   vO 

00   r--i£J  00  10 

t^     TJ-     NN     (50 

^     ^    oo      CO   Tf-    f  1 

CO  ^  in  (>  0 
■+  el          in 

0     fl     ClOO     0 

fo  rooo    to  in 

f  1  lO  10    -^lO 

t^vo    fl    fl    tr> 

CO  fo  in  — 

m       M   CO 

^  in  in  in 

in  m  in  CO  to 

rl    m  fl    fl 

d     c*     «     ft     « 

in  ^  CO 

^     H  oo    OsM    O 

m  o    I'OiO    ca 

-j-  in  r^O    O 

rl  sD   ■+  in  in 
CO  rl    n*  in  n 

On  ON  CO  HI    - 

^  ly-i  r^  Tf  r*i 

NO    M    in  in 

a, 

^  xr,^    t>    ■4- 

rj-  M           m  ^ 

^   in  in        to 

in  rl    **■         in 

-    ^  «    un 

rl    .f  CO  to 

O     •<*•  <0  rt-   ON 

CN  r-   *oo   ■* 

•+   !7-00    in  ■+ 

NO    r^   iniO    M 

ON  r-~  *1-  in  •*• 

in  HI    to  0    to 

-     OnOO   00 

0     t^  C<  VO     CO  o 

in  in  d        00 

O    cl    r^  >♦■  CO 

cl    to  in  M    in 

tl-    ON    0      0      t-. 

ON  CO  0  NO    in 

to    0      t^NO 

I-.    CO  «    to  c) 

**      CO    CO 

M               Cl      fc^      CO 

to  M    CO  ^    rl 

fl            to  »N     « 

fl  -  n 

fl      fl                  M 

5    ^  in  M   o   ^ 

tJ-           CO  CO,  u-» 

00    I    HI    m  0 

0    H    t^iO    0 

l^iO    CO  in  in 

NO     f^OO  00     tj- 

d   f<   «  NO   t^ 

00   in  m  r^ 

HI    CO  rl          HI 

fl      CO    HI 

rl    n    *  •+  rl 

«.    rl    M    in 

in        ro  CO  CO 

th  m  M    M 

oo  ^   fl    ^-.  Ml 

CO  o    in\o    M 

ONfl    rj-  O    0 

0    n    0'  rl    m 

n    to  ^-  t^  « 

HI  00    fl    to  in 

00    NO      fl      0 

1: 

"     ir;         «    UN  to 

-     N     ■>*-   «     CO 

m  rl           CO 

rl    ■+        rl    - 

CO  -f  rl           ro 

«       „      Tf     fl      HI 

fl    in  in  Th 

u-i-sO  VC     t^   Cs 

it*  in  CO  O    H. 

n     CO    CN   cooo 

Oi  ^  in  n    h^ 
rl    if  rl    Oi  •+ 

in  ^  0    ro  in 

Tf  On  HI    rl    « 

0    ON  m  rl 

C.     ^  rl    CO  tr,  vt- 

r--  r-^  in^ 

«    ^  If  t-^  in 

0    OMO  00  00 

0    in  fl  NO    rf 

NO      ^    t^NO                ! 

1-    H          ~ 

"^               M 

HI      rl      fl      M      HI 

fl     to   -4     -■     fl 

CO                  n     n 

** 

1 

ft>  t^  CN  «    ro  p^ 

lO  00    O    invc 

0     O     t^-OO     Oi 

Oi  rl  oo  ^    CO 

r^vc    ^4-  in  On 

H.      4    '»•     cl 

.+  »f  «    0    to 

t'-OO    in  in 

CO    CO    ir-,    CO    CO 

T^  vo  to  rl    in 

CO             »      Tf 

in  CO  fl    -*    HI 

in  in  CO  rl    rl 

•+  ■+  -    rl 

i?  c)    moo    «  ^ 

'O    »n  Cn  rl    O 

o  «"   (^  t~.  t~~ 

to  to  to  t--   rl 

On  in  ON  0     0 

NO      ■-      t^NO      0 

fl     On  ON  t- 

•*•              •>♦•  cl 

rl    r«         «    cl 

Tj-    fl      -,      - 

1^  in  ro  i4-  .f 

to  to  N-          in 

'f  ■+      ~  - 

M    fl    ro  M 

«  t-^  ^•.  cooo   M 

O  inia    t-~  0 

vO    CTi  fl    'f  •♦ 

O     H.   00     -<     - 

0    -  00    I*-  ON 

CO    0\   M      t^OO 

CO  0    0    0 

f«           M 

rl         M   rt 

cl      -      -                  M 

M               MM 

rl    rl 

\     &1 

ififfcTyrM 

i-rc-T'S'rT'S 

x'c^T— "t-^o" 

!i\'i^\:cn-t 

o-jTr-ro-ri-.- 

1^  i^ii  rlni 

■"-"-r'rTi^." 

Cl  Hi  r-c 

— (  I<l  HI          r-1 

S^               H^ 

*4  ff)  »\ 

-I         71  ri  Cl 

-1  — I  -<  Cl 

.c 

.  >.  -•  £;  2 

Cf,  ;u  ^,  1-5 

^  c   =  o  ^ 
X  l-I  1-5  C 1^ 

<  /;  0  ft  r:; 

bi:_;  c  C  - 

<    ^    HIHIO 

>-5  J--J»t- 

CO  00  oc  X  00 

X  C5  O  O  3 

o  Hi  Hi ,—  ri 

iri  -M  •M  C^  CO 

S  «  2  2  3 

-f    -f   -t"    -r   IC 

•p  X   t;.   1- 

lO  i.t>  iC  "0  >o 

iC!  ir?  -c  -x  -^ 

?o  ;c  -^c  «£>  XI 

■£>  (£>  -w  ■£  ■■£> 

:c  --s  -c  X  X 

*  X  X  X 

oo  X  X  a:  00 

X'  X  X  X  X 

X  X  X  X  X 

X  X  •/■/  X  X 

■X  X  X  X  X 

HH    T^    HI 

X  X  X  X  X 

HI             HI 

HH   H- 1    Hi    f'    HH 

HI    HI    HI    H"    HI 

Hi    Hi    rH    Hi    HI 

n    HI                 HI    HI 

! 

«e !  -  /■  St  o 

'-  91  r?  >*  >!? 

•£  1 .  r  «  s 

—  91W  *'9 

:e  1^  :r  Si  e 

-^9l«-t<i9 

etN.XSt© 

!    .2 

^.1+,^^,* 

,,,.,  ,• ,...  ,^ 

>?  ir  >*  "9  ;£ 

^  ^  «  ^  '<^ 

:c  -^  ;£  tt  1- 

1^.  In.  In.  In.  |, 

1.1-.  I'.  I-.  X 

©111 '51*1 'it 

91  91  91  SI  91 

91 9191 91  91 

'51'^I'M^I^I 

9I9I9I919I 

9I919I919I 

91  91  91  91  91 

615 


TABLE  XIX. 

Elements  of  the  Orbits  of  the  Minor  Planets. 


2 

mil 

coSoH 

H  .2 

v* 

orj           V, 

to 

« 

5 

.s  .  .IS 

t;-T!~.a  c 

g.s.sss 

"■'  a 

a  •- 

(iaspar 
.  Hind. 
j  Gaspar 

1  Ga.spar 

Hind. 
Hind. 

03 

i 


»—:/:-  t-i  Ci  CO 


I-  <■,  7.  P3  Q 
—  'U  ■^  '~  '■"' 

c/:,  oj  ^/?  ^  -vj 


g    S    O    P.  ft. 
^,  <  O  <  < 

t^  1^  I-  X  Ci 

'Ti'  *t  't  -r  -r 

Vj  cc  CC  CO  CO 


S  V.  !l^  S  1-5 

o  =  -s  —  ^ 
•  O  O  '(^  .o  o 
CO  'y,  xj  CO  CO 


.  T  c-l  jr. 

rt  r-  N  f  I  I— 

s  <  4  <  -i; 

Cl  CI  C\  C-1  CI 

i*  to  '--J  >r:  -o 

CO  CO  'X   X  CO 


■3 

-J  t-t    (« 

'i      •      •  B    o 

.-3  ^  '^  ;:i.  c; 

^  a  a  ■^-  rt 

C  S  5  C  iJ 

•n  to  .O  ..-5  O 


:  o  o  c  s. 

CI  CI  01  ^^  « 

.-  .o  ..-:  ,,-.  .o 

CO  CO  rx  CO  CO 


h^  -^  h5  r^  H-l 


.  r-  C^ 


lO  CO  I 


J3  ja 

>^  >    J-    U    .^ 
<^  /^  ^,  r^  1-3 

r:  :■■:-*  -f  f 
ic  uT  .-1  .o  .n 

r/j  'X  V:*  ■/)  VD 


c  a  ?;  -5 

a      »;  =  C      4J 

tt-r  i  gJ3 

•  .  _<  .4  ra  — - 


rJ;  C  O  -<'■< 


O  -+  ('I  N 

a^  »-^  M    O  vO 

O  -t-  r»    rt    ►-. 
-rt-  -t-  r<    r-  ►- 

T*-  rt-  -t  «T+ 

6  6  6  6  6 


^    i-<    ONOC  vo 

t^  ro  to  c«^  ^f 

d  d  6  d  6 


CC  oo  *t-  •+  ON 
O  ro  »n  r*^  O 
u^  H-  O  ^  rl 
O^-oc    «    rl    H 

oo  %c  ^  i-t  ri 
m  »*!  -t-  Tt-  ^ 

d  d  6  6  d 


O   O   c^  »/^  rt 

*^  f  1  O  CT^^ 
Os  m  a\  >ri  o\ 
vy,  m  O  r^  •-• 
\£)  C^sC  OO  oc 
rf-  r*^  rn  ro  r*^ 

d  d  d  d  d 


CC  00  l-^  Pi  O 
t^  ^  ro  ly^O 

»n  C^  irv  r*^  rO 
O  "^  Cn  t-.  O 
OC  \0    -"    O^0O 

r*^  -^  -i-  •+  «^ 

d  d  d  d  c 


On      r-. 

0\  T^  ^r\  ^  O 

ro^£3  O  I-"  f* 
r4  "i-oc  rl  (?v 
-t*  O  r^  r~.  ro 
ri   :^  -+  O  t^ 

rt-  C^  rj-  tJ-  c*^ 

d  d  d  d  d 


O  oc  «  \0  00 

oc    -+•  rl  ^    O 

yo  rl  t-.  c^  t~^ 

rj-  -+  -i-  rt-  rt- 

6  6  6  6  6 


00  VO         u-1  O 

11  VCJ  oo  00    Pi 

r»   O   O   c^  O 
O  oc    O  VO  o 

►^  a\  4*  r^  i^- 

I--,  t-^oc  o^oo 


•i-t  O  CO  00    o 

rl  sT;  Os  CNOo 
rl  o  O  oo  I-" 
oo  oc  ro  r^  i-t 
O   VO  ro  m  rn 

ON  pi  -sO  pi  ■4- 

r<i^  oo  ^    pn 

^  o  O  c^so 


£}   rl   M   T^ 
«    t^O    N    O 

p*^  •+  ooo  «o 
to  po  i^  O  *^ 
*-  OC  oc    rl    ■+ 

-^^  4"  t^  f*^  >^ 

rl    C^  'o  »o  pl 

o  CNtc  oo  oo 


ro  00 

p^^o  oc  r--  t-^ 
O   u-1  ^oc  oo 

'^o  \o  ■-<  r^  »o 

t^  O    '-'  M    i^ 

c>  pi  d  d  oc 

O   ^    rl  ro  ^- 


oc 


r*"OC  r^  -rt*  pi 
^  p^X;  ro  to 
VO  N-  o  ^  pi 
lo,  lo  -I   r-oo 

ro  "4-  t*"\d  t*^ 
ro  «  po  po  lo 

O  t^oc  vi    c> 


oc  00  OC    -^-oo 

•sO    -+•  PI     rj-  ro 

-t-  o  n  -+  ^ 
oc  o^  rl  p*^  r-^ 
>o  Cn  ■■  c*^  pi 
CN-o  ^  a^  >n 
I-  oc  so  '^  t~^ 
oc   c^  r-oc  c^ 


00  o  <:>  r*^  -' 
O  M3  'O  «o  *i- 
»o  r~--oo  wioo 

oo  »oo  oc  t^ 
CO  pi  1^  lo  d 
ro  to  ro  O  00 

so  CC  r--oo  ^ 


oc  »ooo  POM      ooHP^^'1-      MCNMr-»-pi       p*^popipit^     sot^oocc^-^      poHv'^t^f*^      p*^  r^*.D  w  r-. 


5;    rl  r^  »o\o  O 
M  rl  p<^  CO 


tJ-  to  rt-  to  O 


^  O  VO    f  1  VO 
PO  VO  to  LO 


i-t     O     0     LO  rf- 


■+  -^  f^  PO  Pi 

to  «+■  -i-  Pi  t*^ 


rl  oc   o^  t*^  t^ 

rt"  to  to  to  ■^ 


poosr-«io      lopi-t-^o 


O   P»   O  O   t 
M    Pl    CO  »■ 


t^  o  ■«*■  r-  ^n 

■<4*  pl   po       i-i 


r^  r^  Pl  ^00 


TJ-OC  00    pl  00 
^  (-.  to  to 

CNC'»'*'PiPo      O'jcr^  Ti-'^o       T*-  po  r^  ON  Pl 

MtOpl-+Tj-  U^,   POP-.I-.  -t"'i"*+  Pl 

o  1^  toso  ■^     to  CTsoo  -^t"  t~-.     p*  •+  c>vo  pi 


ooootooo       O00«*ip» 


-o  Pl  • 

to  Tj- 


to  tooo   O   O 


VO  oc    POsO    Cv 

■^-  Pl  >o  po  -i- 


\0    t^OO    VO  ■+ 
►1     ■•4-  to   W 

r-  ro  o  r--  ^- 

to  rl   to       rf- 


Pl    Pl    OOO    VO       VO  •^\D 


r--oo  »o  to  ON 


f-vo  VO  -^  O" 

VO  t-'      TJ* 


P0\0     ON  pi     »^ 
to  to  r^ 


Ov  r^vo   pi  VO 
-+  P4    »o  to 


VO  to  tooc    ■+ 

Tt-  -^   -I-  to 


_      O    •+  to  t^  VO 


rf*  to  to  to  to        rJ-OC  OOv*-"  rOLoO"^0 


t^  0*vO    OvvO 

^  Pt  Pl  'J- 


^Pl 

CvvO    Pl 


q  q  c^  qvvq 

VO  to  OvOO    t^ 


r-  CNvo  VO 

Pl    pl    to  Pl 


cs 


to     tooc  vonoo       o»t^cor^io    ^opl^^'-•^--     pioopi'.oto 


t--.  tosc    Pl  00 
to  to  rl   11 


O    to  Cv  ► 


rl 


O   Pl   O   to  PH 
OC  r-  r^  O  '+• 


r^  'ooo  L*^  1-* 

w   »o  4*  PO 

ON  r-  r--  »-*  po 

to  ^  w    to  tJ- 

OO     Ov  O  00  vC 
to  to  M  VO  OO 

11      pl       N-  tl 

t-^  M  to  I-I  rl 


^,  *-«  VO  to  ■»!• 
fl  Ti"  lo  pl  ►I 

r-.  -t-  to  f  1  Pl 

to  H<     ^  lo 

to  to  to\o   po 

Pl     ro  -i-00     O 

11    rl  Pl 

-:t-  O    O  00    to 


t^  rl-  Ov  ONO 
•1  ^Pl 


1  to  PI    VO 

rl   tJ- 


VO  Pl 


O  n  "^ 
to  —  O 
^  n  Pl 


t--  ov  r--  Pl  r^ 

to  rj-  w 

1^  VO  O      Pl     LO 

rl  (o  ^  .« 

00  VO  VO    to  M 


to  to  ON  Pl     pl 

ON  li    On  LO  ON 


Os  to  CN  >OOC 
to 


■•i-VO      LO 
to  to  (O 


t    00  VO    n  00  OO 
w    rorl    -^h 


OO    Pl    ■^  On  to 
■+  Pl    VO  -1-  PO 


PO  tJ-  to  O 
Pl    VO         « 


rj-  On  to  Pl    Pl 
w    to         to  LO 


q  O    O    Pl  00 

r^  r^  «  -4-  "-^ 

<1    n    to  to  to 
^sO    -+  to  r^  On 

pi      Pl  LO  Pl 


'i-OO    M    O    Os 

LO   TJ-  -i-  to 

to,  to  -t-sO  oo 

rJ*  C^  't-  t*^  O 

n    to  to 


'^-nvOtoH-        OsOvOOOOv 


t^  O    OvOO    VO 

to  ^00  LO  pi 


to  o 

Pl  to  to  NN      n  cl 

toOvor^O         wOstoplLo      vOn 


to  O  oo 

n  to  ON 


oc  VO  »-  VO  to 
to  Tt"  rl  LO 


to  VO  ^00  Os 


r--oo  to  o  Pl 

tl  »o  Pl  -i-  O 


f  00  VOOO  •- 
CO  •+    ^  to 

H  O  OV  ■+  LO 

CO  rl  00  »o 

Pl  .-.to 

so  "-;  t--  rl  O 

VO  N  VO  ON  ON 


VO  to  vosO  00 
Pl  to  to  VO  pl 

VO    l^  Pl  VO    n 

tooo  Pi  VO,  ro 


P)     M     M      1 

^  rl  PO 


>0 


Ov  PO  -+  rl 

ro  tl 


O  VO    O   H    to 

W    LO  Pl    -^  Pl 

r-.  w   ONVO   tJ- 

CO  M     pi      M      PO 

to       to  n  rl 

o  o  o  o  o 


t^  r^  **•  ■^*  CO 


:?« 


&    1 


G    ^ 


P 

•-:  "-s  /;  t-5  W2 

•lO  tC  »0  *  ^■'i 
<o  to  «o  «— t  ^O 

00  oo  CO  CC  oo 


tOVO  LO  1--S  On 
OO  VO  to  to  0> 
Pl    NM  .1 

o  o  o  o  o 
o  o  i-^  o  c^i 

CO  «  (M 

a  a  G  GXi 

3    rt    «J    3    « 

l-5H5^t-3&M 

«c  o  -y:  CO  -^ 
to  to  "^  tO  «o 

00  ;/:  OD  c/:  CO 


to  On  1    ON  Ml 

M     to   rf-  to 

^  Pl  *f  *^  LO 
M  to  to 

ONVO  O  't*  H 
to^o  Pl  PO  Pl 
Pl         rl    ►-<    n 

0  w  o  o  o 

1  -  o*  cj  od  o 
C^        CI  c^ 


«    3    S  O    S 

to  f-   -^  -*■   '^ 

eo  '-T  '.c  to  '1^ 

c/i  CC  en  ca  <Xi 


O    f- 


q  CO  Pl 

t^  PO  4- 

to       H" 


Os  PO  r-OC        vO    tooo    CO  o 


to  »^  N    O    « 
(^  C^  'I-  LO  O 

H-   to  N*  rl 

O   O   r^  LO  t-- 


to  r-.  O  oc  w 
w  r--oo  Lz-'vo 
n   ^         rl   ^ 

o  .i-i  o  o  lO 


rt    3    rt    3    3 

T-:  ^  ■-:  ^-3 

t^  tc  -r  Ci^  to 

tC  to  i/^  CO  i:© 

CC-  CO  CO  or  CO 


vosO  VO    O    ►-' 
-^  n  H 

O   1^  'i-'+l^ 

pi      PO  M      M     M 

^-  GNOO    O    On 
t^OO  VO    rj-  r-- 
n 

o  o  o  o  o 
c^i  o  o  o  ci 


t-t  <  ^  <  i< 

re  '—  r-  -*  >c 

lO  «0  '-O  «o  «o 

CO  CO   'X   CC  CO 


LO  H     t-^  Pl     LO 

LO  to  pl    to  Pl 

to  t^OO    M    N 
Pl   po 

M      O*    PO   r^-\Q 

>0  rf-  O    O    O 
to  M    to  »M    to 

o  o  =  o  o 


5   ri   rt    ?3   3 
M  'O  (M  «0  'O 

L.O  to  to  to  '-O 

CO  CO  CO  00  QC 


-i-  VO  rl   to 
pl    •*•        po 


00    ■+  O    to  c> 


n  Pl  -4- 1-~  •+• 


flS  ce  ^4>  =  3 

t-3>-2  Ht,  <  ^ 

t^  ifi  CO  '■'^  :C 

CO  '■-:  tc  to  -^ 
CO  CO  CO  oo  x 


to    c4 

^=3 


a  "-.  i: 


c!  *" 


» 


4 

a 


r  =  ^  X 


di 


J    <i 


K'i  .i  -  ^      3  "s  J=  .a 


s 


'i9»«*-»  ~«t-ac*j   222J2   2'-2235   5?{p^-J5  *S;^$SJ   «f;??;?i2 

Bi — ai^— — ^  1  — 1~  '         mil 

C46 


w^."  "J*--,  IW^-" 


TABLE  XIX. 

Elemente  of  the  Orbits  of  the  Minor  Planets. 


5  C  6 

iSS  . 

r  I.  I-  t. 

A      =      "      " 

=    2^3 

r  o  ;^  J 

»  GO  CD  OS 


co<-<i 
-»-  -t  i.*^  o 

>n  1.-5  >o  >^ 

O^  -X  CA)  C*v 


ri  r--  c>  t^ 
6  6  6  6 


^  ri   1^  >-r.  O 

-i  oc  r--oo  ^ 

M    -1-  ir.  -t-  >/^ 

^  rri  t^  a^  r»     II 


o  o  a>  c>^o 


^  Vi-i  C^  i^OC 

>>>  -i-  "i"^  "^ 

■r^  M     >/^  ('^  f^ 

^  00    1^,00    ^ 
ro+        ■4-"-, 

►«    O    O  +  "^ 
ro  rl        oo   >'^ 


vO 


t~.  «   O 


tl    «   -   I 


r»T+  M  O   i-i 

ON  O  'I-  "^  O 
-   CO  NH  ej 

~b  b  f^-  T  "^ 

N    -1-        CO 

00   -i-  O  CO  ;> 


o  +  O  r 
c)  +  I  -  ■ 


<^  ir>  ^^  o  c* 
r-  e-l  CI 

.    .    ■  ^v" 

Kj  w.  ^  <  >-; 
..T  tC  O  -^ 

—  '-"^  '^  S  '-i 

<«  CO  <K  =0  •'5 


o  si.    ■£ 
■%.a>'li  i 

S    o    o  .2    4* 
•-  ?»  f?  "*  '/} 


t3           .     .  '-J 

-a          -va 

—5  ~3 

-3 

a    S|a 

a       a  a 

a  3  c 

.  a 

•gcESg 

•ggcf  ^ 

c  i:  ~  "3  J) 

s-S 

to     U     O      =     OT 

w:    O    C     y,    ;/j 

C     O     A     y.    3 

S'S  §  S2 

^  siuS^ 

^•  "^ 

O    S  J5  J=    = 

O   3    o   o   o 

o  3  =   s  a< 

Ci^oua 

^H  fl^  o^  r«  ^ 

fi,»JCOfi( 

hJ'J! 

a       s  =  a 

■9.  c  u  °  S.  — 

•A    K    ^    C    3  X 

2  5  w  5  '"  2 

"^  3  3  .a  «  "3 

;5  J  J  ^'  Cn  C 


i    :3   01   <y 


.    B 

—  tt 
3   : 


£  a 
^11 


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1-^  »n  M 

r^  tn  c^ 

0  \0  ^3    C^  t-^ 

OOOO    M    fl    o 

f»  wt-  o 

•  +  -H-  H- 


\£)  VO  00  •-<  CO 
iy-1  CO  c^  w  CO 
COOO    C^  '-'    t^ 

\  r\  M   o*'  t^  f*^ 


lo  co^    ro3c 

^  f  J  lio  O  oo  *^ 
'^c<  t-<  rt  ^ 

I    -+  f»   »r^  t^  *r» 

f  \0    ►-    c*^  1--  t*^ 

I  M     M     *i«     CO 


.!-i>« 


TABLE  XXL    Constants,  &c. 


Base  of  Naperian  logaritlinis 
Alodulus  of  the  common  logarithms 
RadiiiH  of  a  Circle  in  seconds 

"         "        II       II  minutes  . 

"        "        "       II  degrees 
Circumference  of  a  Circle  in  seconds 
II  II        II      when  ?■  =  1 

Sine  of  1  second 


.     c  =  2.71828183 
^0  =^=  0.43429448 
.    J- =  2002(i4.80() 
r  —  3437.7468 
.    r  :::=.  57.29578 
1290000 
.   ■"■  =  3.14159205 


log 

0.43429448 
9.03778431  — 10 
5.31442513 
3.53027388 
1.75812203 
0.11200500 
0.49714987 


0.000004848137    4.08557487 


Equatorial  horizontal  parallax  of  the  sun,  according  to 
Encke    .... 


8'^5711C 


0.9330390 


Length  of  the  sidereal  year,  according  to  Hansen  and 

T      ?/"^''!'.  ■        '■    , 365.2503582  days    2.50259778 

l^ength  ot  the  tropical  year,  according  to  Hansen  and 

*^-^'"''*'^" 365.2422008    //        2.50258095 

-  o''4(j()0OOO24  ^'''  ^'"^'^  ""^  '^'  *'''*'''"'  ^''"'  ''  '^"'   ^^^^'^'      '^^^"  '"'"""'  '■"""''""    '' 

Time  occupied  by  the  pa-ssage  of  light  over  a  distance 

equal  to  the  mean  distance  of  the  earth  from  the 

sun,  according  to  Struve 497.»827 

Attractive  force  of  the  sun,  according  to  Gauss         .        /,• :.-  0.017202099 

"  "        "         "  "  "       '/        in  se- 

*=°"^«°f^'"^ 3548.18761  3.55000057 


2.0970785 
8.23558144  — 10 


Constant  of  Aberration,  according  to  Struve 20"  4451 

"         "  Nutation,  //  n  peters "        9' '.2231 

Mean  Obliquity  of  the  ecliptic  for  1750  +  t, 

according  to  Uessel        ....  23°  28' 18".00  — 0".48308<- 0".00000272295<» 
Moan  Obliquity  of  the  ecliptic  for  1800  +  t, 

according  to  Struve  and  I'etors     .        .  23°  27' 54".22  —  0".4738<   —  0".0000014C^ 


General  Precession  for  the  year  1750  +  t,  according  to  Bosscl 
"  "         "  "  II         II  Struve 


50".21129  +  0".00024429G6i 
50".22980  +  0".000226< 


Masses  of  the  Planeto,  the  Mass  of  tue  Sun  being 


THE  UNIT. 


Mercury    , 


.  wi— - 


1 


Veni" 


Fvtth 


Mars     . 


•        •        •        • 


4805751 

1 

390000* 

1 

354936 

1 

2680637' 


Jupiter 


Saturn 


Uranus 


m  — - 


1 


1047.879 


Neptune . 


Oitf 


24905 

1 

18780* 


EXPLANATION  OF  THE  TABLES. 


Table  I.  contains  the  values  of  the  amjlr  of  the  vertical  and  of  the 
logarithm  of  the  eai-th'«  radius,  Avith  the  geographical  latitude  as  the 
argument.  The  adopted  elements  are  those  derived  by  Bossel  De- 
noting by  I,  the  radius  of  the  earth,  by  ^  the  geographieal  latitude, 
and  by  f'  the  geocentric  latitude,  we  have 

/  =  ^  —  1  r  30".65  sin  2<p  +  1".16  sin  4^  —  &c 

log  f.  =  9.9992747  +  0.0007271  cos  2^  -  0.0000018  cos  i^  +  &c., 

(>  being  expressed  in  parts  of  the  equatorial  radius  as  the  unit  These 
quantities  are  required  in  the  determination  of  the  parallax  of  a 
heavenly  body.  The  formuUe  for  the  parallax  in  right  ascension  and 
111  declination  are  given  in  Art.  61. 

Table  II.  gives  the  intervals  of  sidereal  time  corresponding  to 
given  intervals  of  mean  time.  It  is  required  for  the  conversion  of 
mean  solar  into  sidereal  time. 

_  Tabt,e  III.  gives  the  intervals  of  mean  time  corresponding  to 
given  natervals  of  sidereal  time.  It  is  required  for  the  conversion 
of  sidereal  into  mean  solar  time. 

Table  IV.  furnishes  the  numbers  required  in  converting  hours 
minutes,  and   seconds   into   decimals   of  a  dav.     Thus,   to  convert 
Vih  19m  43.5s  into  the  decimal  of  a  day,  we  find  from  the  Talkie 

13/i    =0.-5416667 

19m   ==0.0131944 

43s     =0.0004977 

O.Sa  =  0.0000058 

Therefore  13/i  19m  43.5^  =  0.5553646 

651 


652 


THEORETICAL   ASTRONOMY. 


y//^ 


^< 


Tho  decimal  corresponding  to  0.5s  is  found  from  tliat  for  os  by 
changing  the  place  of  the  decimal  point. 

Tarle  V.  serves  to  find,  for  any  instant,  the  number  of  days  from 
the  beginning  of  the  year.  Tims,  for  1863  Sept.  14,  15/i  53//i  37.2.*?, 
we  have 

Sept.  0.0  =  243.00000  days  from  the  beginning  of  the  year. 
Ud  15h  53m  37.28:^    14.66224 
Required  number  of  days  =  257.66224 

Tarle  VI.  contains  the  values  of  JI/=75  tan  ^v  +  25  tan''  ^v  for 
values  of  v  at  intervals  of  one  minute  from  0°  to  180°.  For  an  ex- 
phmation  of  its  construction  and  use,  see  Articles  22,  27,  29,  41, 
and  72. 

In  the  case  of  parabolic  motion  the  formulae  are 


m  = 


15- 

3 


M=m{t  —  T), 


wherein  log  Cu  =  9.9601277.  From  these,  by  means  of  the  Table,  v 
may  be  found  when  t  —  jT  is  given,  ov  t  —  T  when  v  is  known.  From 
I,  =.  30°  to  V  =^  180°  the  Table  contains  the  values  of  log  M. 

Tarle  VII.,  the  construction  of  which  is  explained  in  Art.  23, 
serves  to  determine,  in  the  case  of  parabolic  motion,  the  true  anomaly 
or  the  time  from  the  perihelion  when  v  approaches  near  to  180°. 
The  formulae  are 


smttJ 


•''/200 


w  +  \ 


t  —  T-- 


200 


sin'  ^v 


w  being  taken  in  the  second  quadrant.  The  Table  gives  the  values 
of  A^  with  ?y  as  the  argument.  As  an  example,  let  it  be  required  to 
find  the  true  anomaly  corresponding  to  the  values  t  —  T=22.5  days 
and  log  q  =  7.902720.     From  these  we  derive 

log  il/=  4.4582302. 

Table  VI.  gives  for  this  value  of  log  M,  taking  into  account  the 
second  difierenccs, 

V  =  168°  59'  32".49 ; 


but,  using  Table  VII.,  we  have 

w-=168°59'29".ll, 


\  =  3".37, 


EXPLANATION  OP   THE  TABLES.  653 

and  Jicnce 

v  =  w-\-A,=:  168°  59'  32".48, 
tlie  two  results  agreeing  completely. 

Table  VIII.  serves  to  find  the  time  from  the  perihelion  in  the 
case  of  parabohe  motion.  For  an  explanation  of  its  constrnction 
and  use,  see  Articles  24,  69,  and  72. 

Table  IX.  is  used  in  the  determination  of  the  true  anomaly  or 

he  tune  from  the  perihelion  in  the  case  of  orbits  of  great  eccen- 

Art.  41  '"°'*''"^*^'^"  i«  f»"y  ^'^Pjained  in  Art.  28,  and  its  use  in 

Table  X.  serves  to  find  the  value  of  .  or  of  <  -  ^  in  the  case  of 
e  hptic  or  hyperbolic  orbits.  The  construction  of  this  Table  is  ex- 
planied  .n  Art.  29.  The  first  part  gives  the  values  of  log  B  and 
l!t  f ,''  t^ie  argument,  for  the  ellipse  and  the  hyperbola. 

Li  the  case  of  og  C  there  are  given  also  logl.Diff.  and  log  half  II. 
Diff.,  expressed  in  units  of  the  seventh  decimal  place,  by  means  of 
which  the  interpolation  is  facilitated.  Thus,  if  we  denote  by  log  (C) 
he  va  ue  which  the  Table  gives  directly  for  the  argument  next  less 
than  the  given  value  of  .1,  and  by  aA  the  difference  between  this 
argument  and  the  given  value  of  .1,  expressed  in  units  of  the  second 
decimal  place,  we  have,  for  the  required  value, 

log  C==  log  (C)  +  A^  X  I.  Diff.  +  A^^  X  half  II.  DifT. 

For  example,  let  it  be  required  to  find  the  value  of  log  (7  correspond- 
ing to  ^  =  0.02497944,  and  the  process  will  be :-  ^ 


Arg.  0.02, 


(1)  (2) 

log  (C)  =  0.0034986      logl.Diff.  =4.24.585  log  half  II.Diff.  =  1.778 

(1)—  8770.6   logA^l     =9.69718  2IogA4  —9304 

A^=  0.497944,         (2)= ^^8  gio^S  J^o 

log  C=  0.0043771 

The  second  part  of  the  Table  gives  the  values  of  ^  correspondinff 
to  given  values  of  r.  x  & 

Table  XI.  serves  to  determine  the  chord  of  the  orbit  when  the 
extreme  radu-vectores  and  the  time  of  describing  the  parabolic  arc 
are  given.  For  an  explanation  of  the  construction  and  use  of  this 
iable,  see  Articles  68,  72,  and  117. 


654  T1IE(»11KTI('AF.    ASTIJONOMY. 

Tablk  XII.  exhibits  the  limits  of  tlio  real  roots  of  the  equation 

sin  (s'  —  Z)  =^  vig  sin*  /. 

The  construction  and  use  of  this  table  arc  fully  explained  in  Articles 
84  and  93. 

Tahles  XIII.  and  XIV.  arc  used  in  iluding  the  ratio  of  the 
sector  included  hy  two  radii-vectorcs  to  the  trianj^h^  included  by  the 
same  radii-vectorcs  and  the  chord  joininj^  their  extremities.  For  an 
explanation  of  the  consti'uction  and  use  of  these  Tables,  sec  Articles 
88,  81),  {);],  and  101. 

Table  XV.  is  used  in  the  determination  of  the  chord  of  the  part 
of  the  orbit  described  in  a  <>;iven  time  in  the  case  of  vcrv  eccentric 
elliptic  motion,  and  in  the  determination  of  the  interval  of  time 
whenever  the  chord  is  known.  For  an  explanation  of  its  construc- 
tion and  use,  see  Articles  116,  117,  and  119. 

Table  XVI.  is  used  in  finding  the  chord  or  the  interval  of  time 
in  the  case  of  hyperbolic  motion.  Sec  Articles  118  and  119  for  an 
explanation  of  tin;  use  of  the  Table,  and  also  the  explanation  of 
Table  X.  for  an  illustration  of  the  use  of  the  columns  lieadcd  log  1. 
Diff.  and  log  half  II.  DiiK 

Table  XVII.  is  used  in  the  computation  of  special  perturbations 
when  the  terms  depending  on  tlu!  squares  and  higher  powers  of  the 
masses  are  taken  into  account.  For  an  explanation  of  its  construc- 
tion and  use,  see  Articles  157,  165,  166,  170,  and  171. 

Table  XVIII.  contains  the  elements  of  the  orbits  of  the  comets 
which  have  been  observed.  These  elements  are:  T,  the  time  of  peri- 
helion passage  (mean  time  at  Greenwich);  /T,  the  longitude  of  the 
perihelion;  Q,,  the  longitude  of  the  ascending  node;  /,  the  inclina- 
tion of  the  orbit  to  the  plane  of  the  eclijitic;  e,  the  eccentricity  of  the 
orbit;  and  g,  the  perihelion  distance.  The  longitudes  for  Xos.  1,  2, 
12,  16,  91,  92,  115,  127,  138,  155, 156, 159,  160,  162,  171, 173-175. 
180,  181,  185,  191,  192,  195-199,  201,  203,  204,  207,  208,  212-215, 
217-219,  221-228,  230,  233,  234,  237-248,  251-258,  261-267, 
269-275,  277-279,  are  in  each  case  measured  from  the  mean  equinox 
of  the  beginning  of  the  year.  In  the  case  of  Nos.  134,  146,  172, 
182,  189,  190,  205,  231,  232,  236,  259,  and  268,  the  longitudes  arc 


KX  PLAN  AT  ION  OF  THE  TAHLKS. 


066 


comets 

[f  peri- 

of  the 

Incliiiii- 

oftlie 

2-215, 

1)1-267, 

^quinox 

0,  172, 

lies  tire 


moasurod  from  the  mean  equinox  of  the  beginninf^  of  the  next  year. 
The  longitudes  for  Xos.  19  and  27  are  measnred  from  the  mean 
c<[iiinox  of  LSoO.O;  for  Xo.  18(1,  from  the  mean  ecjninox  of  .luly  .'5; 
for  No.  187,  from  the  mean  cc^uinox  of  Nov.  9;  for  No.  200,  from 
the  mean  equinox  of  July  1;  for  No.  202,  from  the  mean  ecjtiinox 
of  Oct.  1 ;  for  No.  20G,  from  the  mean  equinox  of  Oct.  7;  fiir  No.  211, 
from  the  mean  e(iuinox  of  1848.0;  for  No.  216,  from  the  mean  ecjui- 
nox  of  Feh.  20;  for  No.  220,  from  the  mean  ecjuinox  of  April  1 ;  for 
No.  250,  fntm  tiie  mean  equinox  of  Oct.  1;  and  for  No.  276,  from 
the  mean  equinox  of  1865  Oct.  4.0. 

No.s.  1,  2,  11,  12,  20,  23,  29,  41,  53,  80,  and  177  give  the  elements 
for  the  successive  appearances  of  Ilalley's  comet;  Nos.  104,  116,  126, 
143,  149,  157,  167,  170,  176,  178,  183,  194,  210,  220,  235,  249,  and 
260,  tliose  for  Encke's  comet,  the  longitude-  being  measured  from  the 
mean  equinox  for  the  instant  of  the  perihelion  passage.  Nos.  92, 
127,  159,  172,  196,  and  222  give  the  elements  for  the  successive  aji- 
pearances  of  liiela's  comet;  Nos,  187,  216,  250,  and  276,  those  for 
Faye's  comet;  Nos.  197  and  238,  those  for  Rrorsen'.s  comet;  Nos. 
217  and  243,  those  for  D' Arrest's  comet;  and  Nos,  145  and  245, 
those  for  Winnecke's  comet.  For  epochs  previous  to  1583  the  dates 
are  given  according  to  the  old  style. 

This  Table  is  useful  for  identifying  a  comet  which  may  appear 
with  one  previously  observed,  by  means  of  a  similarity  of  the  ele- 
ments, its  periodic  character  being  otherwise  unknown  or  at  least  un- 
certain. The  elements  given  are  those  which  appear  to  re[)reseut  the 
observations  most  completely.  For  a  collection  of  elements  by  vari- 
ous computers,  and  also  for  information  in  regard  to  the  observations 
made  and  in  regard  to  the  place  and  manner  of  their  publication, 
consult  Carl's  I-tcpcrtor'min  der  Cometcn- Astronomic  (Munich,  1864), 
or  Galle's  Comctcn-Vcrzcichniss  ap])ended  to  the  latest  edition  of 
Olbers's  Mdhodc  die  Jkihn  eines  Cometcn  zu  berechnen. 

Table  XIX.  contains  the  elements  of  the  orbits  of  the  minor 
planets,  derived  chiefly  from  the  Berliner  Asironomisches  Jahrlmch 
fur  1S6S.  The  epoch  is  gi.'on  hi  Berlin  mean  time;  J/ denotes  the 
mean  anomaly,  (p  the  angle  of  ec'.v>ntricity,  /z  the  mean  daily  motion, 
and  a  the  semi-transverse  ax'^.  The  elements  of  Vesta,  Iris,  Flora, 
Metis,  Victoria,  Eunomia,  Meljiomene,  Lutetia,  Proserpina,  and 
Pomona  arc  mean  elements;  the  others  are  osculating  for  the  epoch. 
The  date  of  the  discovery  of  the  planet,  and  the  name  of  the  dis- 
coverer, arc  also  added. 


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THf:OUETICAL   ASTRONOMY. 


Table  XX.  contains  the  mean  elements  of  the  orbits  of  the 
major  planets,  together  with  the  amount  of  their  variations  during  a 
period  of  one  hundretl  years.  The  epoch  is  expressed  in  Greenwich 
mean  time,  and  L  denotes  the  mean  longitude  of  the  planet. 

Table  XXI.  gives  the  values  of  the  masses  of  the  major  planets, 
and  also  various  constants  which  are  usetl  in  astronomical  calcula- 
tions. 


APPENDIX. 


M=.  171°  36'  10"  +  39".7&  (t  ~  1750), 

-^  =  50".2113  +  0".0002443  (t  -  1750) 

+  (0"  4889  -  0".00000614  (t  -  1750))  cos  (A  -  M)  tan ,.,        (i) 


=  -  (0  ■.4889  -  0-.00000614  (,,  _  1750))  ,i„  (;  _  3/) 


by  ..,  we  have,  according  ,0  I^,,  ''""'' "'  "'"' '"""  ^•«'+  ^. 

-^^- =  0".17926  -  0".0005320786  r, 

-^^  =  50".37572  -  0".000243589  r, 
0  =  23°  28'  18".0  +  0".0000098423  r», 


and  if  we  put 


ai       dt       ^' 


dt 


sim,~±  =  n, 


dtuS  w  c;!"""""'  '"^'"'^ '"  "•^"'  -"-  w  -^ 


41 


(fa 

-^  =  m  -\-  nt&n  d  aia  a, 


dd 

dt 


=  n  cos  o, 


(2) 


067 


658  TIIEORfmCAL  ASTRONOMY. 

and  the  numerical  values  of  m  and  n  are,  for  the  instant  f, 

TO  =  46".02824  +  0".000r,086450  («  —  1750), 
n  =  20",06442  —  0".0000}>70204  (t  —  1750). 

To  determine  the  precession  during  the  interval  t' — <,  we  compute 
the  aniual  variation  for  the  instant  J  (/'+  t)  and  this  variation  mul- 
tiplied hy  t'—  t  furnishes  the  required  result. 

Ji.  Nutation. — The  expressions  for  the  equation  of  the  efpiinoxes 
and  for  the  nutation  of  the  obliquity  of  the  ecliptic  arc,  acconling  to 
Peters, 

A?.  =  — 17".2405 sin  ft  +  0".2073  sin  2^  —  0".2041  m\  2C  +  0".0677  mi  ( C  —  T') 

—  l".2(;i)4  sin  20  +  0".1279  sin  (©  —  r) 

—  0".02i:jHin(O4-r), 

Ae  r=  -f  9".2231  cos  ^  —  0".nS97  cos  2^^  +  0".0S86  cos  2C 

+  0".o510  cos  20  +  0".0093  cos  (©  +  r), 

for  the  year  1800,  and 

AA  =  —  17".2577  sin  SI  -}  0".2073sin  2Q,  ~  0".2041  sin  2C  +  0".0677  sin  (C  —  T') 

—  1  ".2t)9o  sin  20  +  O".127o  sin  (Q  —  r) 

—  0".0213  «in  (O  +  r), 

Ar  =  +  9".2240  cos  Ji  —  0".0896  cos  2$J      (f'.OSSS  COS  2C 

+  0".5507  cos  20  +  0".0092  cos  (Q  +  r), 

for  the  year  1900.  In  these  equations  SI  denotes  the  longitude  of 
the  ascending  node  of  the  moon's  orbit,  referred  to  the  mean  cfjuinox, 
C  tlie  true  longitude  of  the  moon,  0  the  true  longitude  of  the  sun,  F 
the  true  longitude  of  the  sun's  perigee,  and  P  the  true  longitude  of 
the  moon's  perigee.  The  values  of  these  quantities  may  be  derived 
from  the  solar  and  lunar  tables,  and  thus  the  required  values  of  a^ 
and  AS  may  be  found.  The  equations  give  the  corrections  for  the 
reduction  from  the  mean  equinox  to  the  true  equinox. 

To  find  the  nutation  in  right  ascension  and  in  declination,  if  wo 
consider  only  the  terms  of  the  first  order,  we  have 


(4) 


The  values  of  aA  and  Ae  are  found  from  the  preceding  equations,  and 
for  the  differential  coefficients  we  have 


Ao=: 

da 

aA 

-t- 

da 
de 

AC, 

a3  = 

aA 

f 

ds 

(is 

Ae. 

APPENDIX. 


659 


!0f 

vcd 
the 


wo 


(4) 


and 


-J—  =  cos  e  +  8111  £  tan  o  sill  o, 


=  —  cos  a  tan  d, 


^1 


—  !r=  COS  &  Sin  £, 


----  sin  o. 


(5) 


The  tonns  of  the  second  order  arc  of  sensible  niagiiitiide  only  when 
the  body  i.s  very  near  the  pole,  and  in  this  esisc  by  eoiiijuitiiij;  the 
secoru  differential  eoeffieients  the  eoinplete  values  may  be  iouiid. 

Ii  the  reduction  of  the  place  of  a  ])lanet  or  comet  from  the  mean 
eqninox  of  one  date  t  to  the  trne  equinox  of  another  date  /',  the 
determination  of  the  correction  for  precession  and  of  that  for  mitatiop 
niav  be  efl'eetcd  simnltancoiislv.  Thns,  let  r  denote  the  interval 
t' — /  cxju'csscd  in  parts  of  a  ycijr,  and  the  snni  of  the  corrections  for 
f  recession  and  nntation  j^ivcs 

Ao  =  mr  -\-  aA  cos  £  -f~  ('i"  +  ^^-  ^h»  £)  sin  a  tan  <*  —  A£  cos  a  tan  <J, 

-(-  Ar  sin  o.  ^  ^ 


a5  = 
Let  ns  now  put 


(in  -\-  A/,  sill  £ )  cos  o 


mr  -f-  aX  cos  £  =/, 
nr  -\-  A/  sin  e  =  fj  sin  G, 
—  Ae  =  ff  cos  G, 


(7) 


and  the  e<inations  (G)  become 


Aa  =/  -\-  ff  sin  (  C  +  a)  tan  <5, 
A<J=         (jcoa(G-\-a), 


(«) 


as  already  given  in  Art.  40. 

The  astronomical  ei)liomerides  give  at  intervals  of  a  few  days  the 
values  of  the  quantities/,  fj,  and  G  for  the  reduction  of  the  jilace  of 
the  body  from  the  mean  e(iuiiiox  of  the  beginning  of  the  year  to  the 
true  e{piinox  of  the  date;  and,  in  order  to  obtain  uniformity  and 
accuracy,  the  beginning  of  the  year  is  taken  at  the  instant  when  the 
mean  longitude  of  the  sun  is  280°.  When  these  tables  are  not  avail- 
able, the  values  of/,  //,  and  G  may  be  Ibiind  directly  by  means  of 
the  equations  (7).  The  reduction  from  the  true  ecjuinox  of  /'  to  the 
mean  equinox  of  t  will  be  obtained  by  changing  the  signs  of  the 
corrections. 

C.  Aberration. — The  aberration  in  the  case  of  the  planets  and 
comets  may  be  considered  in  three  different  modes: — 

1.  If  we  subtract  from  the  observed  time  the  interval  occupied  by 


660 


TIIEORI-mCAL  ASTRONOMY. 


the  light  in  passing  to  the  earth,  the  re.«ult  will  be  the  time  for  whieh 
the  true  plaee  is  identical  with  the  apparent  place  for  the  observe<l 
time. 

2.  If  we  compute  the  time  occupied  by  light  in  traversing  the 
distance  between  the  body  and  the  earth,  and,  by  moans  of  the  rate 
of  the  variation  of  the  geocentric  spherical  co-urdinates,  compute  the 
motion  during  this  interval,  we  may  derive  the  true  place  at  the  in- 
stant of  observation. 

3.  We  may  consider  the  observed  place  corrected  for  the  aberration 
of  the  fixed  stars  as  the  true  place  at  the  instant  when  the  light  was 
emitted,  but  as  seen  from  the  place  of  the  earth  at  the  instant  of 
observation. 

The  Ibrmula;  for  the  actual  alxirration  of  the  fixed  stars  are — 


^X  =  —  20".44r)l  cos  (X  —  O)  sec  ji  —  0".3429  cos  (I  —  T)  sec  ,3, 
A/9=  4-  20".4451  sm  (?.  —  ©  )  sin  ^  +  0".3429  sin  (A  —  /')  sin  <J), 

in  the  case  of  the  longitude  and  latitude,  and 


(9) 


—  20".4451  (cos  O  cos  e  cos  a  -f-  sin  O  sin  a)  see  3 

—  0".3429  (cos  /'cos  e  cos  a  -\-  sin  /'sin  «)  sec  <5, 

a5  =  -j-  20".44ol  cos  O  (sin  a  sin  >^  cos  e  —  cos  3  sin  e) 

—  20".44r)l  sin  ©  cos  a  sin  S 

-j-  0".3429  cos  /'  (sin  o  sin  3  cos  e  —  cos  d  sin  e) 

—  0".3429gin/'cososin<5, 


ao) 


in  the  case  of  the  right  ascension  and  declination.  In  these  formula) 
/^denotes  the  longitude  of  the  sun's  perigee,  and  they  give  tlic  cor- 
rections for  the  reduction  from  the  true  place  to  the  apj)arfcat  place. 

1).  Intensity  of  LUjht. — If  we  denote  by  r  the  distance  of  a  planet 
or  comet  from  the  sun,  by  J  its  distance  from  the  earth,  and  by  C  a 
constant  quantity  depending  on  the  magnitude  of  the  body  and  on  ita 
capacity  for  reflecting  the  light,  the  intensity  of  the  light  of  the  body 
as  seen  from  the  earth  will  be 


/= 


r»J« 


(11) 


"When  the  constant  C  is  unknown,  we  may  determine  the  relative 
brilliancy  of  the  comet  at  different  times  by  means  of  the  formulii 


J5  = 


(12) 


APPENDIX. 


661 


In  the  ca.sc  of  ti.e  plnnct.s  wc  adopt  as  tlie  unit  of  tho  intensity  of 
lif^l.t  the  vahie  of /when  the  planet  is  in  opposition  nn<l  both  it'nn.l 
tljc  earth  are  at  their  mean  distanei's  from  the  sun.     Thus  we  .)l,tain 


and  hence 


j_  oVa  — 1)» 


(13) 

Let  us  now  denote  by  R  the  ratio  of  the  intensities  of  the  liL'ht 
for  two  consecutive  stelhir  magnitudes;  then,  if  we  denote  by  M  tJie 
apparent  stellar  magnitude  of  tho  planet  when  /- 1,  and  by  m  the 
magnitude  for  any  value  of  /,  we  shall  have 


and  hence 


m  =  J/  — 


R' 


lojr/ 


(H) 


By  means  of  photometric  determinations  of  the  relative  brilliancy 
of  the  stars,  it  has  been  found  that 


and  hence  we  derive 


R  =  2.56, 
j»=3/— 2.45  log/, 


(15) 

by  means  of  whicli  the  apparent  stellar  magnitude  of  a  planet  may 
be  determined,  /  being  found  by  means  of  equation  (13).  The  value 
of  M  must  be  determined  for  each  planet  by  means  of  observwl  val»e«» 
of  w. 

Example.— The  value  of  3/ for  ^un/Homc  is  10.4;  required  the 
apparent  stellar  magnitude  of  the  planet  when   log  a^O'^iTOS 
log  r  =  0.2956,  and  log  J  =  9.9952.  *  ' 

The  equation  (13)  gives 

log /=  0.5129, 
and  from  (15)  we  derive 

wi=:  10.4  — 1.3  =  9.1. 
For  the  values  log  r  =  0.4338,  log  J  =  0.2357,  we  obtain 

log /=  9.7555  — 10, 
and 

TO  =  10.4  +  2.45  X  0.2445  =  11.0. 


662 


THEORETICAL   ASTIIOXOMV. 


E.  Xiimrrh'nf  Cnlnihitionn. — Tho  oxtondcd  numerical  mlfiilatinns 
required  in  many  of  the  prol)lemH  of  Tlicoretieal  Astrononiy,  render 
it  important  that  a  judieion.s  arranj^ement  of  the  details  whould  l)c 
eftected.  The  beginner  will  not,  in  general,  he  able  to  efVcet  such 
an  arrangement  at  the  outlet ;  and  it  would  only  confuse  to  attempt 
to  give  any  specific  directions.  Familiarity  with  the  formuhe  to  be 
applied,  and  practice  in  the  performance  of.  calculations  of  this 
character,  will  speedily  suggest  those  various  devices  of  arrangement 
by  which  skillful  computers  expedite  the  mechanical  part  of  the 
solution.  There  arc,  however,  a  few  general  suggestions  which  may 
be  of  service.  Thus,  it  will  always  facilitate  the  calculation,  when 
several  values  of  a  variable  are  to  be  computed,  to  arrange  it  so  that 
the  values  of  each  function  involved  shall  appear  in  the  same  vorti- 
cal or  horizontal  column.  The  course  of  the  ditferences  will  then 
indicate  the  existence  of  errors  which  might  not  otherwise  be  dis- 
covered until  the  greater  part  if  not  the  entire  calcidation  has  been 
completed;  and,  besides,  by  carrying  along  the  several  i)arts  simulta- 
neously the  use  of  the  logarithmic  and  other  tables  will  Im)  iacilitated. 
Numbers  which  are  to  be  frequently  used  may  be  written  on  slips  of 
paj)er  and  applied  wherever  they  may  be  reijuircd;  and  by  performing 
the  addition  or  subtraction  of  two  lt>garithms  or  of  two  numbei*s  from 
left  to  right  (which  will  be  effected  easily  and  certainly  after  a  little 
jmictice),  the  sum  or  difference  to  be  used  as  the  argument  in  the 
tables  may  be  retained  in  the  memory,  luid  thus  the  required  number 
or  arc  may  be  written  down  directly.  The  luunber  of  the  decimal 
figures  of  the  logarithms  to  be  used  will  depend  on  the  character  of 
the  data  as  well  as  on  the  accuracy  sought  to  be  obtainetl,  and  the  nse 
of  ai)proximate  formula)  will  ha  governed  by  the  same  considerations. 
Whenever  the  formulas  furnish  checks  or  tests  of  the  accuracy  of  the 
numerical  process,  they  should  be  applied;  and  whenever  th'^se  are 
not  provided,  the  use  of  differences  for  the  same  purpose  should  not 
be  overlooked.  By  proper  attention  to  these  suggestions,  much  time 
and  labor  will  be  saved.  The  agreement  of  the  several  proofs  will 
beget  confidence,  relieve  the  mind  from  much  anxiety,  and  thus 
greatly  facilitate  the  progress  of  the  work. 


THE   END. 


